feat(home-dialog): populate home_dialog_list on /mypage/index

Walk-down behavior: each call emits the highest-priority unfired
active dialog; subsequent calls walk to the next-priority entry.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

SVSim.EmulatedEntrypoint/Models/Dtos/Responses/MyPageIndexResponse.cs

@@ -123,7 +123,7 @@ public class MyPageIndexResponse
 
     [JsonPropertyName("home_dialog_list")]
     [Key("home_dialog_list")]
-    public List<object> HomeDialogList { get; set; } = new();
+    public List<Common.HomeDialog> HomeDialogList { get; set; } = new();
 
     // ── Room type in session (Special-format windows) ──────────────────────