fix(ui): draw scroll-to-bottom button in a standalone overlay window

Button drawn in the parent window over the ##chat2-messages child was never
clickable: ImGui resolves g.HoveredWindow to the child for that screen rect, so
ItemHoverable rejects any item submitted in the parent. A top-level Begin/End
window is a sibling in the window list and wins the hit-test for its own rect.
ownerId parameter keeps the window name distinct between the main window and
each pop-out, preventing Begin/End collisions when both render in the same frame.

HellionChat/Ui/ChatLogWindow.cs

@@ -1548,7 +1548,8 @@ public sealed class ChatLogWindow : Window
         Tab tab,
         PayloadHandler handler,
         float childHeight,
-        bool switchedTab
+        bool switchedTab,
+        string ownerId = "main"
     )
     {
         // Capture these before entering the child so we can position the overlay
@@ -1581,7 +1582,7 @@ public sealed class ChatLogWindow : Window
         // (which caused it to drift) and avoids the scrollbar's inner clip rect
         // (which caused it to be cut off on the right edge).
         if (_childScrolledUp)
-            DrawScrollToBottomButtonOverlay(childScreenPos, childWidth, childHeight);
+            DrawScrollToBottomButtonOverlay(childScreenPos, childWidth, childHeight, ownerId);
     }
 
     private void DrawLogNormalStyle(Tab tab, PayloadHandler handler, bool switchedTab)
@@ -1631,33 +1632,59 @@ public sealed class ChatLogWindow : Window
 
     }
 
-    // UI-5: floating jump-to-latest button, drawn in the PARENT window after the
-    // ##chat2-messages child closes. Placing it here prevents two bugs:
-    // 1. Inside the child, SetCursorPos adds to CursorMaxPos.y and inflates
-    //    ContentSize.y / ScrollMaxY every frame, so the button chased its own
-    //    shadow and never landed at a stable position.
-    // 2. GetWindowWidth() inside the child includes the scrollbar column, so the
-    //    button overlapped the scrollbar and was clipped by the inner clip rect.
-    // Screen-space positioning via SetCursorScreenPos is immune to both.
+    // UI-5: floating jump-to-latest button.
+    //
+    // Why a standalone overlay window rather than drawing in the parent?
+    // When this button was drawn in the parent window after the ##chat2-messages
+    // child closed, ImGui's hit-test still resolved g.HoveredWindow to the child
+    // (the child occupies the same screen rect). ItemHoverable then rejected the
+    // button submitted in the parent, so it was visible but never clickable.
+    //
+    // A top-level Begin/End window is a sibling in the window list, not nested
+    // under the child, so it wins the hit-test for its own rect and the button
+    // inside it is fully clickable.
+    //
+    // The ownerId parameter makes the window name unique per calling context so
+    // the main window and each pop-out don't share a single ImGui window entry.
     private void DrawScrollToBottomButtonOverlay(
         Vector2 childScreenPos,
         float childWidth,
-        float childHeight)
+        float childHeight,
+        string ownerId)
     {
         var size           = ImGui.GetFrameHeight();
         var pad            = 8f * ImGuiHelpers.GlobalScale;
         var scrollbarWidth = ImGui.GetStyle().ScrollbarSize;
 
+        // Position confirmed correct in-game: bottom-right of the chat child,
+        // inset by pad, and pulled left of the vertical scrollbar.
         var btnPos = new Vector2(
             childScreenPos.X + childWidth  - scrollbarWidth - size - pad,
             childScreenPos.Y + childHeight -                  size - pad);
 
-        ImGui.SetCursorScreenPos(btnPos);
+        ImGui.SetNextWindowPos(btnPos, ImGuiCond.Always);
+        ImGui.SetNextWindowSize(new Vector2(size, size), ImGuiCond.Always);
 
-        if (ImGuiUtil.IconButton(
-                FontAwesomeIcon.ArrowDown,
-                tooltip: HellionStrings.ChatLog_ScrollToBottom_Tooltip))
-            _scrollToBottomRequested = true;
+        const ImGuiWindowFlags overlayFlags =
+            ImGuiWindowFlags.NoTitleBar
+            | ImGuiWindowFlags.NoResize
+            | ImGuiWindowFlags.NoMove
+            | ImGuiWindowFlags.NoScrollbar
+            | ImGuiWindowFlags.NoScrollWithMouse
+            | ImGuiWindowFlags.NoSavedSettings
+            | ImGuiWindowFlags.NoFocusOnAppearing
+            | ImGuiWindowFlags.NoBackground;
+
+        // End() must be called unconditionally after Begin() — ImGui rule.
+        if (ImGui.Begin($"##scroll-to-bottom-overlay-{ownerId}", overlayFlags))
+        {
+            if (ImGuiUtil.IconButton(
+                    FontAwesomeIcon.ArrowDown,
+                    tooltip: HellionStrings.ChatLog_ScrollToBottom_Tooltip))
+                _scrollToBottomRequested = true;
+        }
+
+        ImGui.End();
     }
 
     private void DrawMessages(

HellionChat/Ui/Popout.cs

@@ -118,7 +118,7 @@ internal class Popout : Window
 
         var handler = ChatLogWindow.HandlerLender.Borrow();
         var logHeight = ImGui.GetContentRegionAvail().Y - inputBarHeight - hintBannerHeight;
-        ChatLogWindow.DrawMessageLog(Tab, handler, logHeight, false);
+        ChatLogWindow.DrawMessageLog(Tab, handler, logHeight, false, Tab.Identifier.ToString());
 
         if (inputEnabled && InputBar != null)
         {