Ulr Manager With Parameters After '?'

I have urls of this general pattern example:

‘?page_id=123’, ‘?page_id=223’, ‘?page_id=456’, etc.

These links are generated through wordpress (running as an application within Yii). I have a router and other configuration set up to correctly integrate Yii with wordpress - thanks to this article! I can instantiate these links if I can prepend a ‘/wp/’, so for example: ‘/wp/?page_id=123’ resolves correctly.

I have been trying to write a rule for the URL manager to correctly rewrite the URL based on a parameter coming after the ‘?’ and I have failed. At the risk of showing too much ignorance, I have tried:

'\?page_id=<id:\d+>' => '/wp/?page_id=<id>',

Is this even doable with the URL manager? If so, what would the expression look like?

Thanks for any help!