Through a bit of research I found a winning solution / hack which I have now implemented for myself.

Basically you just need to get yourself a pair of mens suspenders / braces (any style will do!) and fit them underneath the mattress. You clip one end to the bed-sheet on one side, and the other end to the sheet on the other side, and you’re sorted. For a bonus tightening you can also use the middle fixture from the suspenders to pull down the top of the sheet.

We use 2 pairs of suspenders now to make everything work well (one at the top of the bed, one at the bottom).

See below for how it works for us:

Before:

One end of the suspenders:

After:

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