Second-order asymptotic conjecture for hard-square Betti numbers

Let $\mathcal{R}$ be the feasible region for the hard-square homology. Let $(x,y)$ be any point in the interior of $\mathcal{R}$, and let $n_i,j_i,p_i,q_i$ be sequences satisfying

$$p_i\leq q_i,\qquad p_i\longrightarrow\infty,$$

with

$$\frac{n_i}{p_iq_i}\longrightarrow x,\qquad \frac{j_i}{p_iq_i}\longrightarrow y.$$

Second-order asymptotic conjecture. Then

$$\log\dim H_{j_i}[C(n_i;p_i,q_i)] = n_i\log n_i+C_{x,y}n_i+o(n_i),$$

where $C_{x,y}$ is a constant depending only on $x$ and $y$. This would sharpen the paper's factorial-growth result by identifying the linear second-order term in the logarithm of the Betti number.