Sulanke's conjecture on 3-Narayana distributions of tableau statistics

Let $\operatorname{SYT}(n^3)$ denote the set of standard Young tableaux of rectangular shape $(n,n,n)$, and let $b_{r,s}$ denote the tableau statistics used in the source. Define

$$\st_1=b_{1,2}+b_{2,2}+b_{2,3}-2,\qquad \st_2=b_{1,1}+b_{1,3}+b_{2,3}-1.$$

A statistic has a $3$-Narayana distribution when its distribution on $\operatorname{SYT}(n^3)$ is the corresponding $3$-Narayana distribution.

Sulanke's conjecture. On $\operatorname{SYT}(n^3)$, each of the statistics $\st_1$ and $\st_2$ has a $3$-Narayana distribution.

The statement is attributed to Sulanke. The source proves the assertion for $\st_2$ using a bijection, while the assertion for $\st_1$ remains a conjecture; therefore the combined conjecture remains open.