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__/settings-codex-home.test.js

@@ -1,11 +1,7 @@
 /**
  * @file Tests GET/PUT /api/settings/codex-home and setCodexHome: the shared
  * home-dir guard, persistence to the settings file, the `env_owned` flag, and
- * the AI Router layer being re-written into the NEW home for every codex lane
- * on a router (a `codex -p <name>` against a home without `<name>.config.toml`
- * would fail at start); a home change from another site refused, and a
- * CLAUDE_HOME change naming the lanes whose router profile the new home's
- * store lacks. Everything lives in a temp dir.
+ * a home change from another site refused. Everything lives in a temp dir.
  * @author Nguyễn Ngọc Trí Vĩ <vinnt@smartgift.vn>
  */
 
@@ -25,9 +21,6 @@ const { describe, it, before, after } = require("node:test");
 const assert = require("node:assert/strict");
 const express = require("express");
 
-const routers = require("../lib/ai-routers");
-const lanesLib = require("../lib/lanes");
-const { applyRouterToLane } = require("../lib/router-apply");
 const { envOwned } = require("../lib/settings-env");
 const settingsRouter = require("../routes/settings");
 
@@ -97,27 +90,13 @@ describe("/api/settings/codex-home", () => {
     }
   });
 
-  it("switches, persists, and re-writes every codex lane's router layer into the new home", async () => {
-    const cwd = path.join(ROOT, "lane");
-    fs.mkdirSync(cwd, { recursive: true });
-    const { name } = routers.createRouter({
-      label: "9Router",
-      baseUrl: "http://localhost:20128/v1",
-      apiKey: "sk-secret-value",
-      model: "Highest",
-    });
-    const lane = lanesLib.createLane({ title: "c", cwd, tool: "codex", router_profile: name });
-    applyRouterToLane(lane, name);
-    assert.ok(fs.existsSync(path.join(process.env.CODEX_HOME, `${name}.config.toml`)));
-
+  it("switches and persists", async () => {
     const next = path.join(ROOT, "codex-b");
     fs.mkdirSync(next);
     const res = await request("PUT", "/api/settings/codex-home", { path: next });
     assert.equal(res.status, 200);
     assert.equal(res.body.codex_home, next);
-    assert.deepEqual(res.body.routers, { updated: 1, failed: [] });
     assert.equal(process.env.CODEX_HOME, next);
-    assert.ok(fs.existsSync(path.join(next, `${name}.config.toml`)));
     assert.match(
       fs.readFileSync(process.env.AGENTHUB_SETTINGS_FILE, "utf8"),
       new RegExp(`^CODEX_HOME=${next}$`, "m")
@@ -136,13 +115,4 @@ describe("/api/settings/codex-home", () => {
     assert.equal(res.status, 403);
     assert.notEqual(process.env.CODEX_HOME, next);
   });
-
-  it("a CLAUDE_HOME change names the lanes whose router the new store lacks", async () => {
-    const next = path.join(ROOT, "claude-b");
-    fs.mkdirSync(next);
-    const res = await request("PUT", "/api/settings/claude-home", { path: next });
-    assert.equal(res.status, 200);
-    assert.equal(res.body.routers.missing.length, 1);
-    assert.match(res.body.routers.missing[0], /\(c\): 9router/i);
-  });
 });