How to turn video with spherical, equirectangular projection into an actual "360° video" playable with VLC?

I have a video (MP4/AVC) that is in 2:1 aspect ratio and uses equirectangular projection for a spherical video (often called "360° video").

VLC can play spherical videos since version 3.0.0. Through conversion the video in question has "lost" the required metadata to be recognized as a spherical video with equirectangular projection.

I’m not sure which and what metadata to add to the video, in order to fix the video file, such that it is correctly recognized as a spherical video.

This is what I’ve tried so far, but it didn’t work:

ffmpeg -i input.mp4 -c copy -metadata ProjectionType=equirectangular fixed.mp4

Unfortunatelly, this didn’t fix the issue.

After googling around, I was led to believe, that I may fix the issue with MP4Box off the GPAC project, however I’m not sure where to start. I was led to believe it’s not just fixable with simply adding metadata, instead I’d need to change the container types?

In any event, I’d like a solution that uses ffmpeg and/or MP4Box as those tools are readily available to me.

(this question has been originally posted on unix.SE, but I had no answers or comments there, and I figured it’s more in line with this than using Linux)