next-auth v4 session.update() not working right after signin

Environment

next-auth@^4.24.11:
  version "4.24.11"
  resolved "https://registry.npmjs.org/next-auth/-/next-auth-4.24.11.tgz#16eeb76d37fbc8fe887561b454f8167f490c381f"

  System:
    OS: macOS 14.0
    CPU: (16) x64 Intel(R) Core(TM) i9-9980HK CPU @ 2.40GHz
    Memory: 9.30 GB / 64.00 GB
    Shell: 5.9 - /bin/zsh
  Binaries:
    Node: 20.10.0 - ~/.nvm/versions/node/v20.10.0/bin/node
    Yarn: 1.22.21 - ~/.nvm/versions/node/v20.10.0/bin/yarn
    npm: 10.2.3 - ~/.nvm/versions/node/v20.10.0/bin/npm
  Browsers:
    Safari: 17.0

Reproduction URL

https://github.com/hehehai/next-auth-otp

Describe the issue

Because neither session and loading changed after login.

session.update will bail out on the first conditional because session is null.

https://github.com/nextauthjs/next-auth/blob/46e01af41fc028f22322f225f4f03c0dce069035/packages/next-auth/src/react/index.tsx#L472

How to reproduce

Add

session.update(); // also add const session = useSession();

here
https://github.com/hehehai/next-auth-otp/blob/5eae1afbab3212f352b6fdd2f3530d2de9580188/src/app/login/page.tsx#L33-L35

The update function is not going through. And session not updated.

Expected behavior

The session should be updated.

It seems fixed in v5. So I guess it should also be fixed in v4?
https://github.com/nextauthjs/next-auth/blob/39dd3b92de194c1a835f2d87631f4deb9d9fdf65/packages/next-auth/src/react.tsx#L514

Start with packages/next-auth/src/react/index.tsx around line 472 and reproduce the behavior using src/app/login/page.tsx at lines 33-35 from the linked example. Compare the v4 behavior with the referenced v5 implementation, then verify that session.update() runs after sign-in and that the session is updated.