MCP stdio transport for sub-agents (task tool) closes after ~4 seconds while agent is still running

## Bug Report

### Summary
When a sub-agent is launched via the `task` tool, the MCP stdio transport for all configured MCP servers closes approximately **4 seconds** after the agent's connections are established — while the LLM is still generating its first response turn. Any MCP tool calls made after those 4 seconds fail with `MCP server 'X': Error: Not connected`.

### Environment
- **Copilot CLI version**: 1.0.34
- **OS**: Ubuntu (WSL2)
- **MCP server**: engram (local stdio, SQLite-backed)

### Steps to Reproduce
1. Configure an MCP server in `~/.copilot/config.json` under `mcp`
2. Launch a sub-agent via the `task` tool (e.g., a custom agent)
3. Ask the sub-agent to call any MCP tool

### Observed Behavior
From Copilot CLI logs (`~/.copilot/logs/process-*.log`):

```
T+0ms MCP client for engram connected, took 432ms
T+0ms Started MCP client for engram
T+0ms Successfully started MCP server 'engram'
T+0ms Found 11 MCP tools from server 'engram'
... LLM generating response (agent still running) ...
T+4135ms MCP transport for engram closed ← BUG: transport closes here
T+4135ms MCP client for engram closed
... agent continues running for 20-70 more seconds ...
T+25000ms Agent completes
```

All MCP tool calls in the agent return: `MCP server 'engram': Error: Not connected`

### Expected Behavior
The MCP transport should remain open for the **lifetime of the sub-agent context**, not close after a fixed ~4-second timeout during the LLM generation phase.

### Additional Notes
- The main (orchestrator) agent's MCP connections remain open indefinitely ✅
- Only sub-agents launched via the `task` tool are affected ❌
- The 4-second timeout appears to be a hardcoded connection lifetime for sub-agent MCP transports
- The first MCP tool call occasionally succeeds if made within the 4-second window (race condition)
- Reproduced consistently across multiple restarts

### Log Evidence
```
2026-04-22T08:29:42.945Z [ERROR] Started MCP client for engram
2026-04-22T08:29:42.945Z [INFO] Successfully started MCP server 'engram'
2026-04-22T08:29:42.945Z [INFO] Found 11 MCP tools from server 'engram'
2026-04-22T08:29:46.647Z [ERROR] MCP transport for engram closed <- 3702ms later
2026-04-22T08:29:46.647Z [ERROR] MCP client for engram closed
2026-04-22T08:30:11.381Z [INFO] Agent completed successfully <- 24734ms later
```

Start by tracing the task tool's sub-agent MCP client and transport lifetime, then compare it with the main agent's connections using the reproduction configuration and process logs. Reproduce the roughly four-second disconnect and confirm that the transport stays open through agent completion and later MCP tool calls succeed.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.