Young graph isomorphism conjecture for the $Y(14,3)$ family

Let $Y(g,k)$ denote the Young graph associated with the base $g$ and multiplier $k$. $Y(14,3)$ family conjecture. For all $t\ge 2$ and $n\ge 2$,

$$Y((n^2+n)t+n,n^2t+n-1)\cong Y(14,3).$$

This is an experimentally suggested infinite family of Young graph isomorphisms from the paper's computation of equivalence classes; no proof or resolution is given.