[select][menu] Bug with keyboard navigation

Steps to reproduce

Link to live example: (required)

Steps:

1. Go to https://mui.com/material-ui/
2. Then open Components -> Inputs -> Select
3. Use the Basic Select Component and try to navigate using your keyboard.

Current behavior

When I navigate from the main menu to go to the page where you have the MUI Select Component. Using the keyboard to navigate does not work. It only work after I refresh the page.

Expected behavior

Using the keyboard to navigate on the Select Component must highlight the options when the user navigates using their keyboard and select by pressing enter.

Context

No response

Your environment

System:
OS: macOS 13.6.7
Binaries:
  Node: 18.14.2 - ~/.nvm/versions/node/v18.14.2/bin/node
  npm: 9.5.0 - ~/.nvm/versions/node/v18.14.2/bin/npm
  pnpm: Not Found
Browsers:
  Chrome: Not Found
  Edge: Not Found
  Safari: 16.6
npmPackages:
  @emotion/react:  11.11.4 
  @emotion/styled:  11.11.0 
  @mui/base:  5.0.0-beta.39 
  @mui/core-downloads-tracker:  5.15.13 
  @mui/icons-material:  5.15.13 
  @mui/lab:  5.0.0-alpha.168 
  @mui/material:  5.15.13 
  @mui/private-theming:  5.15.13 
  @mui/styled-engine:  5.15.11 
  @mui/system:  5.15.13 
  @mui/types:  7.2.13 
  @mui/utils:  5.15.13 
  @mui/x-data-grid:  6.19.6 
  @mui/x-date-pickers: ^6.18.4 => 6.19.7 
  @mui/x-date-pickers-pro: ^6.18.4 => 6.19.7 
  @mui/x-license-pro:  6.10.2 
  @types/react:  17.0.80 
  react:  17.0.2 
  react-dom:  17.0.2 
  typescript:  4.9.5

Search keywords: MUI Select, Select Highlight

Reproduce the problem at the MUI documentation site by opening Components → Inputs → Select, then using the Basic Select Component after navigating from the main menu. Trace the Select keyboard-navigation behavior and verify that options highlight and can be selected with Enter without requiring a page refresh.