fix(background-agent): preserve parent model in notifyParentSession

Fixes model switching bug where sisyphus_task with category would
change the main session's model after completion.

- Add parentModel to session.prompt body in notifyParentSession
- Add test verifying model is included when parentModel is defined

src/features/background-agent/manager.test.ts

@@ -675,7 +675,32 @@ describe("LaunchInput.skillContent", () => {
   })
 })
 
-describe("BackgroundManager.notifyParentSession - agent context preservation", () => {
+describe("BackgroundManager.notifyParentSession - model/agent context preservation", () => {
+  test("should include model field in session.prompt when parentModel is defined", async () => {
+    // #given
+    const task: BackgroundTask = {
+      id: "task-with-model",
+      sessionID: "session-child",
+      parentSessionID: "session-parent",
+      parentMessageID: "msg-parent",
+      description: "task with model context",
+      prompt: "test",
+      agent: "explore",
+      status: "completed",
+      startedAt: new Date(),
+      completedAt: new Date(),
+      parentAgent: "Sisyphus",
+      parentModel: { providerID: "anthropic", modelID: "claude-opus-4-5" },
+    }
+
+    // #when
+    const promptBody = buildNotificationPromptBody(task)
+
+    // #then - model MUST be included when parentModel is defined
+    expect(promptBody.model).toEqual({ providerID: "anthropic", modelID: "claude-opus-4-5" })
+    expect(promptBody.agent).toBe("Sisyphus")
+  })
+
   test("should not pass agent field when parentAgent is undefined", async () => {
     // #given
     const task: BackgroundTask = {

src/features/background-agent/manager.ts

@@ -639,12 +639,16 @@ Use \`background_output(task_id="${task.id}")\` to retrieve this result when rea
     }
 
     // Inject notification via session.prompt with noReply
+    // Preserve parent session's model/agent context to prevent model switching
     try {
       await this.client.session.prompt({
         path: { id: task.parentSessionID },
         body: {
-          noReply: !allComplete,  // Silent unless all complete
+          noReply: !allComplete,
-          agent: task.parentAgent,
+          ...(task.parentAgent !== undefined ? { agent: task.parentAgent } : {}),
+          ...(task.parentModel?.providerID && task.parentModel?.modelID
+            ? { model: { providerID: task.parentModel.providerID, modelID: task.parentModel.modelID } }
+            : {}),
           parts: [{ type: "text", text: notification }],
         },
       })