Making internal rest requests non-blocking?

I’m making a call to an internal endpoint in order to try and speed up a task.

I don’t need to wait for the task (that awaits at that endpoint) to complete for my script to move on, but it’s still waiting to finish. I’m using the following:

$argsForRequest = array(
'agency_code' => '1234',
'another_param_here' => $myVar,
'timeout'   => 0.01,
'blocking'  => false,
'sslverify' => false,

$request = new WP_REST_Request( 'POST', '/my/custom/endpoint' );
$request->set_query_params( $argsForRequest );
$response = rest_do_request( $request );

Any idea on what I could be doing wrong?