Embeddability conjecture for skeleta of triangulated complex projective spaces

Let $d$ be a positive integer, and let $K$ be a triangulation of the complex projective space $d53f P^d$. Its $d$-skeleton is the subcomplex consisting of all simplices of $K$ of dimension at most $d$.

Embeddability conjecture. If $d=2^k-1$ for some $k\geq 1$, then the $d$-skeleton of $K$ is embeddable into $\mathbb{R}^{2d}$.

This conjecture concerns the exceptional dimensions not ruled out by the Stiefel–Whitney-class obstruction to embedding the $d$-skeleton into $\mathbb{R}^{2d}$. The paper establishes related embeddability results for smooth triangulations, but the stated claim remains open here.