[Fargate] [BUG]: Container status does not properly reflect the actual status

(Because AWS support admitted this is a bug of Fargate, I'm adding this issue for visibility.)

Sometimes a container status doesn't become STOPPED while the actual process (PID 1) in the container exited.
They say it is because the Fargate agent failed to propagate the change of task status to some internal system.

When this problem happens, users must terminate the task manually to remove it.

Reproduces in ECS Fargate 1.4.0.

This roadmap issue names ECS Fargate 1.4.0 and a mismatch between the container's PID 1 and reported STOPPED status; there are no repository files or tests to inspect. Begin by reviewing the reported reproduction and the Fargate task/container status behavior, and consider the issue complete only when AWS addresses the status propagation failure.

Written by the indexing model from the issue text.