401 error for requests to coder.example.com after "Update" option following template change

I am seeing an issue where when I use the "Update" option in the coder dashboard after a template change, the coder process get stuck in this state, where it gets 401 errors trying to talk to the coder agent URL. The issue is resolved by using the "Restart" option instead.

```
2024-06-12 19:43:20.920 [info] connecting to coderd
2024-06-12 19:43:20.934 [warn] run exited with error ...
error= GET https://coder.example.com/api/v2/workspaceagents/me/rpc?version=2.1: unexpected status code 401: unexpected non-JSON response "": Try logging in using 'coder login'.
Error: no response body
```

I am using "token" authentication with the coder agent, with config passed to the container for the auth as:
```
{
name = "CODER_AGENT_TOKEN"
value = try(coder_agent.main[0].token, "")
},
{
name = "CODER_AGENT_URL"
value = data.coder_workspace.me.access_url
},
```

This issue does away after doing a "restart" of the instance.

Discord discussion here: https://discord.com/channels/747933592273027093/1250538421789790271/1250538421789790271

No repository files or tests are named. Reproduce a template change followed by the dashboard's Update option with token authentication, compare it with Restart, and inspect the agent and coderd logs; done means Update reconnects without 401 errors.

Written by the indexing model from the issue text.