Cyclic Young graph conjecture for the family in $Z_{2n-1}$

Let $Y(g,k)$ be a Young graph, and let $Z_m$ denote the equivalence class of cyclic Young graphs on $m$ nodes. Cyclic Young graph conjecture. For $t\ge 3$ and $n\ge 2$, the Young graph

$$Y((n^2+n)t-n^2+n+1, n^2t-(n-1)^2)$$

is in $Z_{2n-1}$. This is an experimentally suggested structural pattern for cyclic Young graphs; the source gives no proof or resolution.