Using query variable as a parameter in a nested field

I'm trying to use a variable passed as a parameter for a nested field.
I can't figure out if its lighthouse not properly parsing this or if I'm using the wrong format.
Is this supposed to work? I inspected the parsed query from the server and it seems I'm getting no value for my varaible $page

import { gql } from "@apollo/client";

export const MY_QUERY = gql`
  query MY_QUERY($group: String!, $page: Int!) {
    myGroups(input: {group: $group, page: $page}) {
      
      groups {
        styleA(page: $page) {
          name
        }
      }

    }
  }
`;


const {loading: Loading, data : Data, error: Error } = useQuery(MY_QUERY, {
  variables: {
    group: group,
    page: page
  }
});

Start with the supplied MY_QUERY and the useQuery variables, then trace how Lighthouse parses variables used in the nested styleA field. No repository file, test, or entry point is named; done means reproducing the query behavior and establishing whether nested-field variables are supported or need a documented fix.