Hard to tell from the video, but it almost looks to me like the issue is Draw Order / rendering order, and not that the attachments themselves are "disappearing" or getting unattached. It looks like when the guy's beard disappears that you can still see a couple pixels of it on the sides of his face, as though the beard is behind his face. Can you confirm this? You could just move the beard to the side, and then see if it is actually disappearing (getting un-attached) or if it is a Draw Order / rendering order issue.
If it is a Draw Order / rendering order issue I would first make 100% sure that you aren't keying the Draw Order in your animation to cause this.
Once you've confirmed that it isn't the Draw Order being set - how are you setting up your characters in Unity? Are you using SkeletonAnimation? Is your project using a true 2D setup (orthographic camera, etc), or do you have any 3D elements going on (perspective camera)?
What is your projects "Transparency Sort Mode" set to? Is it set to "Custom Axis, with Transparency Sort Axis x=0, y=1, z=0? AKA how does your project decide what is rendered in front of other things?
Are you using any special things like Skeleton Render Separator? Are you using Z-Spacing on your SkeletonAnimation? Does enabling the "Fix Draw Order" option on the Advanced section of the SkeletonAnimation fix it?
Taking a screenshot of your SkeletonAnimation with the Advanced tab open might help
