uploaded videos do not play in Safari (on Mac or iOS)

This is a very esoteric (I think) question, and I haven't had any luck at all in hours of googling. I'll try to lay it out as carefully as I can, and if you can offer advice, or a solution, that would be great. I understand this might not be in the scope of your general support!

The simple version of the problem is this. I have a large multisite install. From time to time we have users who upload mp4 or m4v videos to individual sites. Those videos then play fine using the embedded player. That is, they play fine in Chrome and Firefox. In Safari (and Safari in iOS) they don't play at all.

This is WordPress multisite 4.5.3 (and it happens in multiple subsites, with multiple different themes and plugins--and with a default theme and no plugins). It's an older install which has gone through many upgrades, so it still uses the blogs.dir structure for uploaded files.

All other kinds of uploaded files (images, pdfs, doc, etc) work fine in all browsers.

You can see the problem here:
(it's not complete content yet, but just as a test page for now).

That post has the same video twice. The first one it's linked to the video uploaded via ftp to the httpdocs root of our site. The second it's in the blogs.dir directory uploaded via the WordPress video uploader.

As you'll see, the first example plays fine in all browsers. The second example does not play in Safari (and gives a "error loading this reource" error. Not very informative).

Same server. Same file.

Any thoughts or ideas about this? (or happy to offer more information if it will help).