Noncongruent quasi-prime matrices in prime dimensions

For matrices $A$ and $B$ of size $m\times n$, say that they are congruent if there are invertible square matrices $U$ and $V$ of orders $m$ and $n$, respectively, such that $UA=BV$. A quasi-prime matrix is the product of two quasi-invertible semi-prime matrices, where a semi-prime matrix is a rectangular matrix whose smaller dimension is prime. The quasi-prime noncongruence conjecture. For every natural number $k$, there are infinitely many prime numbers $n$ and $(n-2k)\times n$ quasi-prime matrices that are not congruent to each other.

This is presented as a weak form of the preceding conjecture: it retains the existence of infinitely many noncongruent matrices but replaces the semi-prime and quasi-invertibility assertion with quasi-primality. The supplied context gives no evidence that it has been resolved.