feat(coach): deep-link the Coach page into a conversation

The Coach page opened a blank new chat regardless of where the user came
from. Make it a deep-link target: a `?c=<id>` query param opens that
thread on mount, and absent it the page auto-opens the user's most-recent
conversation. "Most recent" is the `updatedAt desc` head of the rail
list, so the same thread opens on web and mobile.

The shared conversation surface grows two additive props
(`initialConversationId`, `autoOpenMostRecent`) that feed its existing
selection state; the history drawer's selection stays the single source of
truth. The drawer surface is unchanged — both props default off and the
rail list is only fetched when auto-open is requested.

src/app/coach/page.tsx

@@ -1,7 +1,7 @@
 "use client";
 
-import { useEffect } from "react";
-import { useRouter } from "next/navigation";
+import { Suspense, useEffect } from "react";
+import { useRouter, useSearchParams } from "next/navigation";
 
 import { CoachConversation } from "@/components/insights/coach-panel/coach-conversation";
 import { useCoachLaunch } from "@/lib/insights/coach-launch-context";
@@ -40,7 +40,37 @@ import { useDisableCoach } from "@/hooks/use-disable-coach";
  * constant, centred, max-width-capped column across the new-chat hero and
  * the active conversation. All of that lives in `<CoachConversation>`'s
  * page branch, so the page stays a thin gating + sizing wrapper.
+ *
+ * v1.18.11 (W11, #67) — the page is now a deep-link target INTO a
+ * conversation. A `?c=<id>` query param opens that specific thread on
+ * mount; absent it, the page auto-opens the user's most-recent thread
+ * (server-authoritative `updatedAt desc` head of the rail, so it lands the
+ * same thread on web + mobile). Both feed `<CoachConversation>`'s existing
+ * selection state — the dashboard Coach entry navigates straight into the
+ * live conversation instead of a blank landing. `?c=new` (or no thread at
+ * all) keeps the new-chat hero. The search-param read sits in a Suspense
+ * child so the client-bailout never de-opts the route.
  */
+function CoachPageBody() {
+  const searchParams = useSearchParams();
+  // `?c=new` is an explicit "start a fresh chat" escape hatch; any other
+  // value is treated as a conversation id to open. A blank/absent param
+  // falls through to most-recent auto-open.
+  const rawC = searchParams.get("c");
+  const deepLinkedId = rawC && rawC !== "new" ? rawC : null;
+
+  return (
+    <CoachConversation
+      surface="page"
+      autoFocusComposer
+      initialConversationId={deepLinkedId}
+      // Only auto-resolve most-recent when the entry did not pin a thread
+      // and did not ask for a fresh chat.
+      autoOpenMostRecent={deepLinkedId === null && rawC !== "new"}
+    />
+  );
+}
+
 export default function CoachPage() {
   const router = useRouter();
   const launch = useCoachLaunch();
@@ -88,7 +118,11 @@ export default function CoachPage() {
       // is hidden, so the page reclaims the full height below the TopBar.
       className="bg-background -mx-4 -mt-6 -mb-20 flex h-[calc(100dvh-8rem-env(safe-area-inset-bottom,0px))] min-h-[32rem] flex-col overflow-hidden md:-mx-6 md:h-[calc(100dvh-4rem)]"
     >
-      <CoachConversation surface="page" autoFocusComposer />
+      {/* `useSearchParams` requires a Suspense boundary so the client-search
+          bailout never opts the whole route out of static optimisation. */}
+      <Suspense fallback={null}>
+        <CoachPageBody />
+      </Suspense>
     </div>
   );
 }