SageMaker remote access broken on VS Code 1.110.0 - "No IDE windows found. Shutting down detached server."

## Problem
SageMaker remote access via deep link ("Open in VS Code") stopped working after
upgrading VS Code from 1.109.x to 1.110.0.

## Root Cause
The detached server (server.js) starts correctly, writes the port to
`sagemaker-local-server-info.json`, but then immediately shuts itself down
with "No IDE windows found. Shutting down detached server." before SSH
can connect to it.

## Steps to Reproduce
1. Upgrade VS Code to 1.110.0
2. Click "Open in VS Code" from SageMaker Studio
3. Connection fails with HTTP status 000 on localhost

## Evidence from sagemaker-local-server.out.log
```
Detached server listening on http://127.0.0.1:XXXXX (pid: XXXXX)
Parent IDE type: vscode
Writing local endpoint info to .../sagemaker-local-server-info.json
No IDE windows found. Shutting down detached server.
```

## Environment
- AWS Toolkit: 3.97.0
- VS Code **1.110.0** (released 2026-03-04): ❌ BROKEN
- VS Code **1.109.5**: ✅ WORKS
- OS: macOS darwin arm64 25.3.0
- Confirmed regression: downgrading back to 1.109.x fixes the issue immediately

## Workaround
Downgrade VS Code to 1.109.1 and set `"update.mode": "none"` in settings.json

Start with server.js and the detached-server shutdown path described in sagemaker-local-server.out.log, then inspect how sagemaker-local-server-info.json is written and how IDE windows are detected. Reproduce with VS Code 1.110.0 and compare against 1.109.5; done means the SageMaker deep link keeps the detached server available for SSH and the localhost connection succeeds.

Written by the indexing model from the issue text.