Healthcheck fail fast feature during container start up

[tporeba]

I have identical question as posted here Healthcheck fail fast feature during container start up · Issue #6751 · docker/docs · GitHub (closed because it was asked on wrong github repo I guess).
From original post:

@petr-ujezdsky https://github.com/docker/docker.github.io/issues/6751

Current HEALTHCHECK command does not allow some sort of fail fast functionality.

Let’s assume some big web app that typically starts up in about 10 minutes. The following config seems to be your suggestion given the documentation:

Dockerfile:

HEALTHCHECK --interval=10s --timeout=3s --start-period=10m\
  CMD curl -f http://localhost/ || exit 1

Now imagine that due to some configuration error the app deploy fails 15s after the docker is run. The web server (Tomcat, Jboss etc.) continue to run though, so the docker container stays up.
Then I would have to wait approx. 10 minutes to know that something went wrong.

I can imagine helathcheck scripts that could be written to detect some early unrecoverable errors like service unavailable 503 (still deploying, there is still hope) vs 404 (not deployed, and probably won’t be). I just don’t see how this additional info can be passed via current healthcheck API which only allows for 0 (healthy) or 1 (unhealthy).

So is there any way to achieve this in current healthchecks?
If no, then are there any chances for this being accepted as feature request ?