I found a very interesting problem in the Open Problem Garden, which I am surprised is not as well-known as I would think it would be:

Prove that If $p>3$ is prime and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb Z_p}$, then there are column vectors $x,y \in {\mathbb Z_p}^n$ which have no coordinates equal to zero such that $Ax=y$.

It is easy to show that this conjecture is false when $p=2,3$. Also, when $p$ is a little larger than $n$, this conjecture has been proven true. However, the case when $n$ is large relative to $p$ seems not only difficult to prove, but also it seems to require nontrivial computational resources to even get a sense of what is going on.

Has any work been done to try to computationally understand this problem? In particular, is this conjecture known to be true for say $p=5$, $n=15$?