apigatewayv2 websocket- put authorizer context variable in $connect route when using IAM authorizer

Hello, I'm trying to use apigatewayv2 websocket.

Is there a way to save context variables at route:$connect when using Iam authorizer? My purpose is verify context variables in $connect route and continue using these variables in any other routes. (To restrict important variables like identityId...)

I know I could do this when using lambda authorizer, with lambda output {context: <any>}. But when I'm trying to use Iam authorizer, I could not find how to do it.