Mills–Robbins–Rumsey's monotone-triangle correspondence conjecture

Let $\TSPP{n}^{k}$ be the subset of triangular shifted plane partitions whose entries in the first $n-1-k$ columns attain their maximal values $n$. Let $\MT{n}^{k}$ be the set of monotone triangles whose entries in the first $n-k$ columns attain their minimum values $j-i+1$. Mills–Robbins–Rumsey's monotone-triangle correspondence conjecture. For $n\geq2$ and $k=0,1,\dots,n-1$,

$$\lvert\TSPP{n}^{k}\rvert=\lvert\MT{n}^{k}\rvert.$$

This proposes an equality between restricted TSSCPPs and restricted monotone triangles, extending the known correspondence between alternating sign matrices and monotone triangles.