Local dimension of tight 39-point codes in octonionic projective 2-space

Let ${\mathbb O}{\mathbb P}^2$ be octonionic projective $2$-space, and let $F_4$ act by its isometries. Let the space of tight codes modulo $F_4$ be the local moduli space obtained by quotienting tight $39$-point codes by this action.

Local dimension conjecture. In a neighborhood of the explicitly constructed code, the space of tight $39$-point codes modulo the action of $F_4$ is a manifold of dimension $24$.

The code is explicitly constructed and a four-dimensional deformation family is exhibited, but the asserted full local manifold dimension $24$ is based on numerical evidence and remains open.