[VS Code] Code generation (streaming) not working with VS Code version 1.97
### Summary
VS Code introduced a breaking change in the last version ([released Thursday last week](https://github.com/microsoft/vscode/releases/tag/1.97.0)). Code suggestions streaming stopped working in this version.
We introduced a hot fix to resolve https://gitlab.com/gitlab-com/gl-infra/production/-/issues/19268+: https://gitlab.com/gitlab-org/gitlab-vscode-extension/-/merge_requests/2371+
Now this issue represents an effort to find and fix the root cause of the issue.
### Steps to reproduce
1) update vscode to be 1.97.0
2) ensure access token has access to code suggestions
3) create new file try to get a code suggestion
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### What is the current _bug_ behavior?

### What is the expected _correct_ behavior?
Code suggestion generated
### Relevant logs and/or screenshots
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### Possible fixes
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issue