Verify canary release

• I verified that the issue exists in the latest Next.js canary release

Provide environment information

Operating System:
  Platform: darwin
  Arch: arm64
  Version: Darwin Kernel Version 22.2.0: Fri Nov 11 02:06:26 PST 2022; root:xnu-8792.61.2~4/RELEASE_ARM64_T8112
Binaries:
  Node: 18.12.1
  npm: 8.19.2
  Yarn: 1.22.19
  pnpm: 7.24.3
Relevant packages:
  next: 13.1.2
  eslint-config-next: 13.1.2
  react: 18.2.0
  react-dom: 18.2.0

Which area(s) of Next.js are affected? (leave empty if unsure)

No response

Link to the code that reproduces this issue

https://stackblitz.com/edit/vercel-next-js-laxqe1?file=next.config.js,components/hello-world/style.module.css,components/hello-world/index.tsx,app/%5Bslug%5D/page.tsx

To Reproduce

1. Create a component with an import to a css modules file.

2. No matter what the context (in a page.tsx, etc...), load this component with await import()

Describe the Bug

The styles aren't injected in the page

Note: if we use a classic import, of course, styles are injected.

Expected Behavior

The styles should be injected in the page

Which browser are you using? (if relevant)

No response

How are you deploying your application? (if relevant)

No response