FIX: hides redundant chat icon on mobile chat routes (#35015)

### Chat in Mobile view

In order to reduce visual noise in mobile view, there is no need to show
chat icon in header, once user has already chat open.

|Before|After|
|---|---|
|<img width="456" height="199" alt="Screenshot 2025-10-01 at 17 16 29"
src="https://github.com/user-attachments/assets/6d124a79-4a0a-4edd-913a-43a0cfa05623"
/>|<img width="442" height="198" alt="Screenshot 2025-10-01 at 17 16 06"
src="https://github.com/user-attachments/assets/052a2f78-574e-4d35-aff4-264f54f2fb15"
/>|

plugins/chat/test/javascripts/components/chat-header-icon-test.gjs

@@ -1,4 +1,5 @@
-import { render } from "@ember/test-helpers";
+import { tracked } from "@glimmer/tracking";
+import { render, settled } from "@ember/test-helpers";
 import { module, test } from "qunit";
 import sinon from "sinon";
 import { forceMobile } from "discourse/lib/mobile";
@@ -43,16 +44,30 @@ module("Discourse Chat | Component | chat-header-icon", function (hooks) {
   });
 
   test("mobile", async function (assert) {
+    const testState = new (class {
+      @tracked isActive = false;
+    })();
+
     forceMobile();
 
-    await render(<template><Icon /></template>);
+    await render(
+      <template><Icon @isActive={{testState.isActive}} /></template>
+    );
 
     assert
       .dom(".icon.btn-flat")
       .hasAttribute("title", i18n("chat.title_capitalized"))
       .hasAttribute("href", "/chat");
 
-    assert.dom(".d-icon-d-chat").exists();
+    assert
+      .dom(".d-icon-d-chat")
+      .exists("chat icon is rendered if chat is inactive");
+
+    testState.isActive = true;
+    await settled();
+    assert
+      .dom(".d-icon-d-chat")
+      .doesNotExist("chat icon is not rendered if chat is active");
   });
 
   test("full page - with unread", async function (assert) {