The refined monotone-triangle enumeration conjecture for cyclically symmetric plane partitions

Let $\CSPP{n}^{k}$ be the restricted class of cyclically symmetric plane partitions, let $\overline U_r(c)$ be its statistic for $1\leq r\leq n$, and define

$$M_n^k(t)=\sum_{m\in\MT{n}^{k}}t^{m_{n,n}-1},$$

where $\MT{n}^{k}$ is the restricted set of monotone triangles. The refined monotone-triangle enumeration conjecture. For $n\geq1$, $1\leq r\leq n$, and $k=0,1,\dots,n-1$,

$$\sum_{c\in\CSPP{n}^{k}}t^{\overline U_r(c)}=M_n^k(t).$$

The paper notes that the left-hand side is later shown to be independent of $r$; the conjecture refines the cardinality comparison between restricted plane partitions and monotone triangles.