Pass along custom query parameter in reset password link

We have followed the instructions here: to pass along a custom query parameter. This lets us use a custom logo and background image depending on that parameter.

Is it possible for us to pass this custom query parameter forward with our reset password link? It currently does not include it which means that the user sees our default page which is a bit of a weird experience to me.

The flow is currently: User attempts to login on login page with custom parameter “Bob”. They see the Bob login experience. User needs to reset their password. User opens the password reset link and instead sees the background and image of our default app instead of our customized login experience

I believe I am going to have to customize the password reset page. I would like the styling to be consistent with the universal login. Is the styling for that page available for us to use? Regardless, I need the custom query parameter to be passed along in the link to make decisions about what is displayed. Is this possible?