[Bug]: Metamask not injected into window object when First time Page load

### Describe the bug

On linux machine when I open my website and I checked is wallet initialised then I am getting false but when i refresh page then its initialized =true

### Steps to reproduce

1.Open website connect wallet
2.close the website and reopen the website and try to connect nothing happened

checked **window. ethereum._state.initialized= false**

### Error messages or log output

_No response_

### Version

10.12.1

### Build type

_No response_

### Browser

Chrome

### Operating system

Linux

### Hardware wallet

_No response_

### Additional context

_No response_

Start by reproducing the first-load behavior in Chrome on Linux with MetaMask 10.12.1, checking whether window.ethereum is injected and initialized before attempting to connect. Done means a newly reopened website can detect the wallet and connect without requiring a page refresh.

Written by the indexing model from the issue text.