Can not work with res.flushHeaders();

[childrentime]

Link to the code that reproduces this issue

https://github.com/childrentime/nextjs-earlyhints

To Reproduce

1. Start next app in dev.
2. check localhost:3000

Current vs. Expected behavior

current: don't have any response

expected: have the correct response

Provide environment information

Operating System:
  Platform: darwin
  Arch: arm64
  Version: Darwin Kernel Version 23.1.0: Mon Oct  9 21:32:52 PDT 2023; root:xnu-10002.41.9~7/RELEASE_ARM64_T8122
  Available memory (MB): 16384
  Available CPU cores: 8
Binaries:
  Node: 20.12.0
  npm: 10.5.0
  Yarn: 1.22.21
  pnpm: 8.15.5
Relevant Packages:
  next: 14.2.4 // Latest available version is detected (14.2.4).
  eslint-config-next: N/A
  react: 18.3.1
  react-dom: 18.3.1
  typescript: 5.4.5
Next.js Config:
  output: N/A

Which area(s) are affected? (Select all that apply)

Pages Router

Which stage(s) are affected? (Select all that apply)

next dev (local)

Additional context

No response