Client won't reconnect on page refresh

### Link to Reproduction
1. Open inspect element and see devtools client connects.
2. Refresh page, keeping the developer tools open.
3. Observe the extension will no longer connect.
4. Close and reopen developer tools and observe devtools reconnects.

Using Firefox Developer version

### `@apollo/client` version
5.11.10

### Apollo Client Devtools version
6.18.14

Reproduce the refresh sequence in Firefox Developer Edition with the Apollo Client Devtools extension while keeping developer tools open. Start by tracing the devtools client reconnect path, then verify that the client reconnects after a page refresh without closing and reopening developer tools.

Written by the indexing model from the issue text.