feat: AI Router per session; Settings at the sidebar's foot; shipped tool icons

AI Router is a session's choice, not a workspace's:
- an AI Router select beside the model picker in the workspace home and a
  pane's run form (RouterPicker.tsx; hidden off Claude/Codex or with only
  Native; last pick remembered in agenthub.session.router)
- `router` on POST /api/lanes/:id/start and POST /api/run; Claude gets
  --settings=<data dir>/router-settings/<name>.json (0600, outranks project
  and user settings, key never in argv), Codex -p <name> + key via tmux -e
- dashboard_runs.router; a resume naming none goes back to it
- profile PATCH rewrites the session files, DELETE removes them
- gone: POST /api/lanes/:id/router/apply, LaneRouterRow, the Add-workspace
  router radio, the claude-/codex-home `routers` answers; clearLaneRouters
  takes the old keys out of each workspace's settings.local.json at start

Sidebar: Settings pinned to the foot, the collapse toggle beside it (stacked
on the rail) instead of in the brand's corner.

Icons: shipped SVG marks for claude, codex, openclaw and hermes (a
black/white pair swapped by theme) in client/public/ai-tools/, used by
AiToolIcon before the server's favicon.

server/__tests__/lanes-api.test.js

@@ -278,65 +278,21 @@ describe("/api/lanes", () => {
     ws.close();
   });
 
-  it("broadcasts lane_update on router apply", async () => {
-    // Without this the client's cached row keeps the OLD router_profile: the
-    // picker looks right until the user switches lanes and back, at which
-    // point the effect re-reads the stale row and flips the selection.
-    const cwd = fs.mkdtempSync(path.join(os.tmpdir(), "lane-router-bcast-"));
-    const created = await request("POST", "/api/lanes", { cwd, title: "Router Lane" });
-    const id = created.body.lane.id;
-    // Start somewhere other than native, so the broadcast has to carry a
-    // CHANGED value rather than the one the row already held.
-    require("../lib/lanes").setRouterProfile(id, "some-router");
-
-    const ws = new WebSocket(`ws://127.0.0.1:${server.address().port}/ws`);
-    const messages = [];
-    await new Promise((resolve, reject) => {
-      ws.on("open", resolve);
-      ws.on("error", reject);
-    });
-    ws.on("message", (msg) => {
-      try {
-        messages.push(JSON.parse(msg));
-      } catch {
-        /* parse error */
-      }
-    });
-
-    try {
-      const applied = await request("POST", `/api/lanes/${id}/router/apply`, { profile: "native" });
-      assert.equal(applied.status, 200);
-      await new Promise((resolve) => setTimeout(resolve, 100));
-
-      const msg = messages.find((m) => m.type === "lane_update" && m.data.lane?.id === id);
-      assert.ok(msg, "router apply should broadcast lane_update");
-      // `native` is stored as null — what matters is that the pushed row is the
-      // post-apply one, not the "some-router" the client still has cached.
-      assert.equal(msg.data.lane.router_profile, null);
-    } finally {
-      // A socket left open by a failed assertion keeps the whole run alive.
-      ws.close();
-    }
-
-    await request("DELETE", `/api/lanes/${id}`);
-    fs.rmSync(cwd, { recursive: true, force: true });
-  });
-
-  it("router apply asks for a restart only while the lane's pane is live", async () => {
+  it("starts a session on the router it names, and refuses one the store lacks", async () => {
     const tmux = require("../lib/tmux");
-    const cwd = fs.mkdtempSync(path.join(os.tmpdir(), "lane-router-live-"));
+    const cwd = fs.mkdtempSync(path.join(os.tmpdir(), "lane-router-start-"));
     const id = (await request("POST", "/api/lanes", { cwd })).body.lane.id;
     try {
-      for (const live of [false, true]) {
-        tmux.__setExecImpl((args) => {
-          if (args[0] === "has-session" && !live) throw new Error("no such session");
-          return "";
-        });
-        const r = await request("POST", `/api/lanes/${id}/router/apply`, { profile: "native" });
-        assert.equal(r.status, 200);
-        assert.equal(r.body.restart_required, live);
-        assert.equal(r.body.run_id, live ? `agenthub-lane-${id}` : null);
-      }
+      tmux.__setExecImpl((args) => {
+        if (args[0] === "has-session") throw new Error("no such session");
+        return "";
+      });
+      const bad = await request("POST", `/api/lanes/${id}/start`, { router: "nope" });
+      assert.equal(bad.status, 400);
+      assert.equal(bad.body.error.code, "EBADROUTER");
+      // The per-workspace route is gone: a router belongs to a session now.
+      const gone = await request("POST", `/api/lanes/${id}/router/apply`, { profile: "native" });
+      assert.equal(gone.status, 404);
     } finally {
       tmux.__reset();
       fs.rmSync(cwd, { recursive: true, force: true });