Local dimension conjecture for 12- and 13-point quaternionic simplices

Let ${\mathbb H}{\mathbb P}^2$ be quaternionic projective $2$-space, and let the space of tight simplices mean the local moduli space of tight simplices, with the stated dimension understood in a neighborhood of the simplex.

Local dimension conjecture. There exists a $12$-point, respectively $13$-point, tight simplex in ${\mathbb H}{\mathbb P}^2$ such that, in a neighborhood of it, the space of tight simplices has dimension $9$, respectively $6$.

The conjecture is motivated by numerical rank-deficiency evidence after existence of the relevant simplices had been established. The local manifold structure and dimensions remain open in the source.