The perfect-game existence conjecture for PackIt! grids

Let $m \leq n$ be positive integers. Write $P(m,n)$ for the set of primes relevant to the $m \times n$ grid, let $\gamma(m,n)$ denote its gap, and let $K(m,n)$ be the total number of turns in a perfect game; write $\mathbf{1}_{K_p}$ for the indicator that $K(m,n)+1$ is prime. Perfect-game existence conjecture. If

$$|P(m,n)| \leq \gamma(m,n) \leq K(m,n)-|P(m,n)|-\mathbf{1}_{K_p},$$

then it is possible to complete a perfect game of PackIt! for the $m \times n$ grid. The preceding bounds show that this is precisely a range in which perfect packings are not ruled out a priori; the authors report no counterexamples in this range, but the assertion remains unproved.