Conjecture on the weight of maximal $(s,ms-1,ms+1)$-core partitions

Let $s$ and $m$ be positive integers, and let $t$ be determined by either $s=2t-1$ or $s=2t-2$. A maximal $(s,ms-1,ms+1)$-core partition is a core partition maximal under inclusion, and its size is its weight. The maximal-core weight conjecture. The size of a maximal $(s,ms-1,ms+1)$-core is

$$\frac{m^2t(t-1)(t^2-t+1)}{6}$$

if $s=2t-1$, and

$$\frac{m^2(t-1)^2(t^2-2t+3)}{6}-\frac{m(t-1)^2}{2}$$

if $s=2t-2$. There are two such maximal partitions, one corresponding to ${\mathcal L}(s)$ and one to its conjugate. This conjecture extends the agreement, when $m=1$, between the weight of a maximal $(s,ms-1,ms+1)$-core partition and the longest such core; the cited prior results concern the case $m=1$, while the general assertion remains unresolved in the supplied text.