Activities are not triggering any challenges

Yes, exactly. That is why I want him to edit that one which we know is one that should count. If he has been using Zwift he might have edited a Zwift ride to make it count.

... on the other hand, the activity is 7 hrs long, and there is a long ~4 hrs gap in the middle. Normally it should not matter, but I wonder where it comes from, and whether it is not a sign of some other problem.

Yes, I know about that gap, but that's not the point, because I have made rides like this in the past and always counted. I use Strava and on Strava it counts. It is only one example of activity, but I have a bunch of other activities with no pauses for anything and no one is counting, you can just check in my profile; nor counting for rides or for running. My activities is not counting for challenges anymore, none of them.
My profile: https://connect.garmin.com/ modern/ profile/ 87b8f833-e0ef-4699-838d-5124e2b22312
(get rid of spaces between / to check)

I checked the JSON data of activity you posted, and it shows the following:

{
"isOriginal": false,
"deviceApplicationInstallationId": 612376,
"agentApplicationInstallationId": null,
"agentString": null,
"fileFormat": {
"formatId": 1,
"formatKey": "tcx"
},

Note the leading isOriginal: false - that means that the activity was either imported from a 3rd party, or somehow heavily modified, and that's likely the reason for not been counted. If it was recorded by a Garmin device, export it as "Original", delete it from Garmin Connect, and then reimport back from the file you exported. 

Unless the file is corrupted or comes from an unknown source, it should get counted. If you need to make some edits, do them one at a time, and observe when the badge (or the distance from the challenge) disappears.

If does not help, contact the Customer Support and send them the file for investigation.

Was that one of the changes Garmin made in January to stop Zwift (and probably other services) from counting for challenges?

I did a quick test and my TCX file is not counting for challenges, but the same activity as a FIT file is. The same behaviour for standard badges. TCX no longer works. It did before.

I think the problem started to happen since last updates because everything worked fine before it. I use this watch since 2019 January and before that the former owner used it maybe since 10 years ago or so. Always worked fine I have plenty badges but suddenly it stoped working. I don't use any other system to record my activities and looks like this watch only uses tcx format, and it automatically imports activities on Strava (wich as I said works fine counting distances). I checked and distance is counting on my gear, quite strange huh?
I tried deleting and synchronize a couple of activities but no luck.
Well, looks like I won't have the right to participate to any challenges just because I have an old equipment. That's sad.
Actually I am angry with Garmin since I got this watch because they won't make an installer for Linux and I have to create a virtual machine only for synchronizing my activities and that's a shame. The whole process to synchronize one activity takes me about 30 minutes and I'm tired of this. I think that challenge "bug" (or whatever that could be called) is the final reason for me to getting rid of my Garmin and go for some other equipment.

Anyway, thank you guys for your answers and for trying to help me.

All newer watches from Garmin now syncs with the mobile apps so there is not much need to use a computer now. Map updates still need a computer.