Conjecture on nonempty parameter sets for PPT entangled edge states

For each integer $n\ge 3$, let $(\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n)$ be the parameters used in the paper's construction of PPT states in $M_n\otimes M_n$, and let the resulting state have bi-rank $(n^2-1,n^2-2n+3)$. Parameter-space conjecture. For every $n\ge 3$, the open set of parameter tuples that produce PPT entangled edge states of bi-rank $(n^2-1,n^2-2n+3)$ is nonempty. Numerical checks establish the construction for $3\le n\le 1000$, while nonemptiness for every $n\ge 3$ is left open.