MP4 video with missing header

[abolibibelot]

A video I downloaded from Dailymotion years ago (2009) is unreadable because of a missing header (the first non empty byte is at offset 2670, then it would seem to be a valid index). I don't know how it ended up like this (was it like this on the server ? was it corrupted by whatever utility was used to download it ? did it get corrupted later on ? the last one seems unlikely as corruption would typically affect a whole cluster), and that particular video is no longer available online. It's not very important as it can be found elsewhere in better quality (it's supposed to be the Jackson 5's performance at the Motown 25th Anniversary show), but still, I'd be curious to know if something like this is fixable, and how. I know that there are tools which can repair MP4 files when a chunk is missing at the end, but with no valid header, common tools won't even try to parse it as a valid video and audio stream. I tried to copy & paste the header of another video downloaded from Dailymotion around the same time — didn't work. Tried to convert it with ffmpeg — didn't work. Tried to convert the edited video with the copied header — to no avail. MKVToolNix fails to open it as well. So does anyone have a deep enough understanding of the MP4 format to know how to reconstruct a valid header, based on available information and educated guesses ?

(The extension is .h264 but videos I downloaded from Dailymotion at that time always had that extension, even though they were in fact MP4 files with video and audio. Although it's possible that this one is actually a H.264 elementary stream, but an elementary stream wouldn't have an index if I'm not mistaken.)
(The file couldn't get uploaded as-is, got an error message saying “invalid file”... no sh#t... so had to zip it.)