Uniqueness conjecture for the 16-vertex triangulation of the real projective 4-space

Let $\mathbb{R}P^4_{16}$ denote the constructed $16$-vertex triangulation of the real projective $4$-space $\mathbb{R}P^4$. Uniqueness conjecture. $\mathbb{R}P^4_{16}$ is the unique $16$-vertex triangulation of $\mathbb{R}P^4$. This would establish vertex-minimality and uniqueness for the triangulation constructed in the paper; the source presents the claim as a belief rather than a proved result.