feat(apes-sidebar): Phase 2 — per-tab state（tasks showInactive 持久化）

- useTasksPrefs：ctx.settings.register('apesSidebar.tasks') namespace，
  uSES 观察 + update 写回，applies:'live' 跨 reload 持久化
- tasks 面板加 showInactive 开关：OFF 时只渲染 running 会话
- filterTasksRows 纯函数导出（hook 需渲染，过滤决策单测锁定）
- locales：tasks.showInactive zh/en
- tests/tabs.spec.ts：+4 个 per-tab state 过滤测试（25→29）
- AGENTS.md：Phase 2 更新（per-tab state 落地，scratch 冒烟 +
  apps·msgs·profile feed 延后——better-sidebar 无对应数据源 + 无 dsh CLI）

门禁：typecheck rc=0 · build rc=0 · test 29/29

tests/tabs.spec.ts

@@ -16,7 +16,7 @@
  * `registerTab` (returns `() => void`, invoked by cordis on fiber teardown).
  */
 import { describe, expect, it } from 'vitest'
-import { apply } from '../src/client/index.tsx'
+import { apply, filterTasksRows } from '../src/client/index.tsx'
 
 /** The five tab ids this plugin owns (the + menu contract). */
 const EXPECTED_IDS = ['apes:tasks', 'apes:agents', 'apes:apps', 'apes:messages', 'apes:profile']
@@ -232,3 +232,36 @@ describe('apes-sidebar tab badges (Phase 2)', () => {
     }
   })
 })
+
+/**
+ * Phase 2 per-tab state: the apes:tasks `showInactive` filter. When the user
+ * toggles the preference off, only running sessions render. This is a pure
+ * decision function (the hook's uSES↔settings wiring needs a render, but the
+ * filter itself is plain and unit-testable).
+ */
+describe('apes:tasks showInactive per-tab state filter (Phase 2)', () => {
+  const sessions = [
+    { id: 'a', running: true },
+    { id: 'b', running: false },
+    { id: 'c', running: true },
+    { id: 'd' }, // running undefined → treated as not running
+  ]
+
+  it('showInactive=true renders every session', () => {
+    expect(filterTasksRows(sessions, true)).toHaveLength(4)
+  })
+
+  it('showInactive=false renders only running sessions (2 of 4)', () => {
+    const rows = filterTasksRows(sessions, false)
+    expect(rows).toHaveLength(2)
+    expect(rows.map((s) => s.id)).toEqual(['a', 'c'])
+  })
+
+  it('showInactive=false on an all-running list keeps everything', () => {
+    expect(filterTasksRows([{ running: true }, { running: true }], false)).toHaveLength(2)
+  })
+
+  it('showInactive=false on an all-inactive list yields empty', () => {
+    expect(filterTasksRows([{ running: false }, {}], false)).toHaveLength(0)
+  })
+})