Sorry, I had missed this reply.
Failures are sent. The code for updating the build status and the stage status can be found here, and this code is shared by the docker runner and the kubernetes runner. There is no difference in runtime behavior, other than kubernetes writing the updated variables to /run/drone/env (the underlying code can be found here).
I ran a quick test and can see the file is being updated with the expected status. Is it possible your cluster is slow to update the file?