Weak positivity conjecture for q-hit numbers

Let $B\subseteq[m]\times[n]$ be a board, let $r$ be a rank, and let $q$ be a prime power. The $q$-hit number $H_r(B,q)$ is defined for this board and rank. Weak positivity conjecture. Given any board $B\subseteq[m]\times[n]$, rank $r$, and prime power $q$, the $q$-hit number $H_r(B,q)$ is nonnegative. This is proposed as a necessary precondition for an affirmative answer to the question of finding a combinatorial interpretation of the $q$-hit numbers; the source presents it as open.