HTTP/2 Push API for workers

A HTTP/2 Push API in a worker will allow to push resources as soon as the client request comes in,
currently the only way to push a resource from a worker is to add a “Link” header to the response, but this means that I can only push once I have the response from origin, (or respond right away and stream the origin’s response when it comes back, but I’d loose cookies and won’t be able to handle error responses).
An API to push directly from a worker would allow me to make use of the time until the origin responds.
While the workers API is designed around the service worker API, an H2 API can be standalone similar to the cache API.