Thank you for the samples, I clearly see the issue when playing the TestFINAL.mkv sample. It happens because the fragments don't start with a keyframe, which is causing the whole trouble. Next time please ensure you start the cut at a keyframe (I-frame) only.
You can still rescue the TestFINAL.mkv video with the following script, which deletes unusable parts of the video:
#PY <- Needed to identify #
adm = Avidemux()
adm.addSegment(0, 0, 3167000)
adm.addSegment(0, 3501000, 2783000)
adm.addSegment(0, 6618000, 2783000)
adm.addSegment(0, 9735000, 2783000)
adm.addSegment(0, 12852000, 3784000)
adm.markerA = 0
adm.markerB = 15300000
Load TestFINAL.mkv in Avidemux, then open a text file containing the lines above via File --> Project Script --> Run Project, then save the resulting video in copy mode.
I was wrong telling you to delete the whole group of pictures at the end of each fragment video. Cutting away just the last one or two frames (and all the frames till the first I-frame of the next appended fragment) is enough. The problem was actually with the start of each fragment, not with the end.