Unresolved part of Sulanke’s three-row conjecture

Prove that the statistic \(\st_1=b_{1,2}+b_{2,2}+b_{2,3}-2\) has a \(3\)-Narayana distribution on \(\SYT(n^3)\).

Background

Sulanke conjectured that two refined statistics on three-row rectangular standard Young tableaux have the $3$-Narayana distribution. The paper proves the assertion for $\st_2=b_{1,1}+b_{1,3}+b_{2,3}-1$ using the involution $_1$, reducing the unresolved portion to $\st_1$.

The requested result is therefore specifically the remaining first-statistic case; the second-statistic part of Sulanke’s conjecture is already established in the paper and is not included as an open problem.