Vite not transpiling imports in TypeScript worker imported using new Worker.

[segevfiner]

Describe the bug

When importing a TypeScript worker using new Worker without {type: 'module'}, Vite will not transpile imports in the TypeScript module (Although it does inject importScripts("/@vite/env"), instead of an ESM import) which results in:

Uncaught SyntaxError: Cannot use import statement outside a module (at worker.ts?type=classic&worker_file:2:1)

Reproduction

https://stackblitz.com/edit/vitejs-vite-bthmfa?file=src%2Fmain.ts

Steps to reproduce

Open the reproduction and look at the console.

System Info

System:
    OS: Linux 5.0 undefined
    CPU: (8) x64 Intel(R) Core(TM) i9-9880H CPU @ 2.30GHz
    Memory: 0 Bytes / 0 Bytes
    Shell: 1.0 - /bin/jsh
  Binaries:
    Node: 18.18.0 - /usr/local/bin/node
    Yarn: 1.22.19 - /usr/local/bin/yarn
    npm: 9.4.2 - /usr/local/bin/npm
    pnpm: 8.6.12 - /usr/local/bin/pnpm
  npmPackages:
    @vitejs/plugin-vue: ^4.4.0 => 4.4.0 
    vite: ^5.0.0-beta.4 => 5.0.0-beta.4

Used Package Manager

npm

Logs

No response

Validations

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• The provided reproduction is a minimal reproducible example of the bug.

[sapphi-red]

Duplicate of #8470