[BUG] Stale "connector requires authentication" reminder persists after successful MCP tool calls (plugin-bundled connectors, desktop app)

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

When a plugin bundles its own local declaration of an MCP connector that is also separately installed and already authenticated as an org/global-level connector, the model's context receives a persistent system-reminder claiming the connector "requires authentication before its tools can be used." This warning appears before any tool call, appears again after a tool call to that exact connector succeeds and returns a valid payload, and continues to reappear on every subsequent turn — even though the call completes normally with no auth error. Observed in two separate surfaces inside the Claude desktop app: a Cowork session, and a Claude Code session run from inside the desktop app — same pattern in both. Does not reproduce in plain Claude.ai web Chat.

### What Should Happen?

The "requires authentication" signal surfaced to the model should reflect the actual, current auth/connection state of the connector instance that will actually serve tool calls. If a call against that connector succeeds, the warning should not have appeared as blocking guidance beforehand, and should not persist afterward.

### Error Messages/Logs

```shell
The following MCP servers require authentication before their tools can be used:
plugin::

This session is non-interactive, so Claude cannot run the OAuth flow here. Tell the user that these servers need to be authorized — for claude.ai connectors, via their claude.ai connector settings; for other servers, via `claude mcp` or /mcp in an interactive session — and that the capability is unavailable until they do. Do not ask the user for authorization codes, tokens, or callback URLs.
```

### Steps to Reproduce

1. Install/enable a plugin that declares its own reference to an MCP connector (`plugin::`), where that same connector is also already connected and authenticated at the org/global level.
2. In a session in the Claude desktop app — reproduced in both a Cowork session and a Claude Code session — invoke a skill/tool from that plugin that calls a tool on the connector.
3. Observe a system-reminder stating the connector "requires authentication" appears before the call is made.
4. Call the tool anyway. Observe it resolves to a working tool id (a different, runtime-resolved connector UID, distinct from the plugin's own declared reference) and returns a normal, successful payload — no auth error.
5. Observe the identical "requires authentication" reminder reappears on the next turn, and the turn after, regardless of the successful call.
6. Repeated in a second, separate session (Claude Code inside the desktop app rather than Cowork) with a different plugin/connector family — same pattern: reminder before the call, call succeeds, reminder persists after.

### Claude Model

Not sure / Multiple models

### Is this a regression?

I don't know

### Last Working Version

_No response_

### Claude Code Version

N/A — Claude desktop app (Cowork mode), not Claude Code CLI

### Platform

Anthropic API

### Operating System

Windows

### Terminal/Shell

Other

### Additional Information

Note: this is a Claude desktop app bug (reproduced in both Cowork mode and a Claude Code session run from inside the desktop app), not a standalone Claude Code CLI bug. Filing here since existing related reports (#59854, #3779) have been filed and triaged in this repo/claude-plugins-official. Several required fields (Claude Code Version, Terminal/Shell) don't cleanly apply here — answered as best as possible.

Start by reproducing the reminder in a Claude desktop app Cowork session and in a Claude Code session inside the desktop app, using a plugin-bundled MCP connector that is also authenticated at the org/global level. Compare the plugin connector reference with the runtime-resolved connector UID and trace when the system-reminder is generated before and after a successful tool call. Done means the reminder reflects the active connector state and does not persist after a successful call.

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