Asymptotic sharpness of the characteristic bound for Heisenberg-group modules

Let $H_1$ be the Heisenberg group, and let $p$ and $q$ denote primes. A module is understood to be over a field of characteristic $p$, and conditions (1) and (2) refer to the conditions in the paper's main theorem.

Asymptotic sharpness conjecture. There exists a prime $q$ such that, for every prime $p \geq q$, there exists a $\frac{p+1}{2}$-dimensional module for $H_1$ over a field of characteristic $p$ which does not satisfy at least one of conditions (1) and (2) of the main theorem.

This conjecture asserts that the hypothesis $p \geq 2d$ is asymptotically sharp. The paper also proposes the weaker formulation that such modules exist for arbitrarily large primes $p$; no resolution is supplied here.