Rank-one factor existence for matrices of psd rank at most k

Let $M\in\mathcal M^{p\times q}_{r,k}$ be a matrix of psd rank at most $k$, and write a psd factorization as

$$M_{ij}=\langle A_i,B_j\rangle.$$

Rank-one factor abundance conjecture. Every such matrix has a psd factorization in which either at least $k$ matrices $A_i$ and $k-1$ matrices $B_j$ are rank one, or at least $k-1$ matrices $A_i$ and $k$ matrices $B_j$ are rank one. The conjecture is proposed as a route to a semialgebraic description of $\mathcal M^{p\times q}_{r,k}$; no resolution is supplied.