I had heard of adinkras but didn't realize that they were meant to play this role. Nor did I realize that the representation theory of supersymmetry is mathematically underdeveloped.

This is fascinating, although it sounds a bit hard to believe. There are thousands of people who for for decades on topics related to supersymmetry, but still the fundamentals are still relatively unknown? I think one problem is that representation theory works for groups and Lie algebra, but supersymmetry is neither.

It's still an algebra and thus a representation theory is defined. Maybe not a convenient one. If you're interested in the subject, have a look on Virasoro algebras.

I (really) am not an expert on this, but to hear that the representation theory of SUSY spacetime is underdeveloped does not a priori surprise me as - from my limited understanding - this probably involves non-compact groups, and the representation theory for e.g. general non-compact Lie groups is immensely harder than for "nice" (e.g. compact) Lie groups and is still underdeveloped within pure mathematics!

In particular, the irreps generally become infinite-dimensional, there is no natural inner product between irreps which could be used in some analogue of Schur orthogonality relations (which are extremely useful in the study of representations of compact or finite groups), etc. The irreps for non-compact Lie groups are built up case-by-case and the general theory is still unknown - in stark contrast to the representation theory of compact Lie groups which is well understood. Adinkras, as used in the paper, are probably one of the case-by-case constructions (I did not read the paper yet, but this is my guess from what you quoted).

Mathematicians have been studying Lie groups for over a hundred years and they still don't have a general representation theory for non-compact Lie groups. What makes you think that physicists could so easily crack this in a mere few decades?

P.S. I work in physics, but I took some courses in group theory at the mathematics department at our university and that is the source of my information.

I feel at times that publications in this field tend to have a high ratio of repeated conjectures and computational ideas over hard results. For a reading experience closer to the usual practice in mathematical physics I recommend the articles by Yang Zhang