Local dimension conjecture for 24- and 25-point octonionic simplices

Let ${\mathbb O}{\mathbb P}^2$ be octonionic 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 $24$-point, respectively $25$-point, tight simplex in ${\mathbb O}{\mathbb P}^2$ such that, in a neighborhood of it, the space of tight simplices has dimension $56$, respectively $49$.

The corresponding simplices are proved to exist in the paper, while the larger local dimensions are inferred from numerical evidence and remain conjectural.