I think https://www.algolia.com/doc/api-reference/widgets/configure/js/ may help, but it’s not something we have added to our templates by default.
For example I configured a default taxonomy category using code like that below.
let term_data = JSON.parse(queried_terms);
let item_parts = []
for ( const item of term_data.terms ) {
item_parts.push( 'taxonomies.product_cat:"' + item + '"' );
}
let the_query = item_parts.join(' AND ');
search.addWidgets([
instantsearch.widgets.configure({
filters: the_query,
}),
]);
The queried_terms
portion came from some inlined data for current WooCommerce category page page.
I suspect it could be adapted and conditionally set based on post type. Worth tinkering with.