Unimodality conjecture for refined plane-partition Betti numbers

For integers $n\ge 0$ and $\delta\in\frac12\mathbb Z$, let

$$b_\delta(\ell,n)=\#\{\pi\in\mathrm{PP}: w(\pi)=n,\; w_-(\pi)-w_+(\pi)+\delta w_0(\pi)=\ell\},$$

where $\mathrm{PP}$ is the set of plane partitions and $w_-$, $w_0$, and $w_+$ are the sums of the entries strictly below, on, and strictly above the diagonal, respectively. Unimodality conjecture for refined plane-partition Betti numbers. The sequences

$$\bigl(b_0(m,n)\bigr)_{m\in\mathbb Z}$$

are unimodal for all integers $n\ge 0$. The result is known computationally for all $n\le 30$, while the paper also establishes parity unimodality; the full unimodality remains conjectural.