A nowhere-zero point in a linear mapping conjecture

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 $$p=5$$, $$n=15$$?