The END-tile parametrization conjecture

Fix $n$ and $p$, and let an END tile be a valid tile with in-carry $c_{\mathrm{in}}=\mathrm{END}$. Let $\lambda$ be a partition with at most $n-2-c$ parts, and let $\mu$ be its conjugate partition with at most $c$ parts, where $c=c_{\mathrm{out}}$.

END-tile parametrization conjecture. The END tiles are precisely

$$(c,\mathrm{END},(p-2-\mu_c,\ldots,p-2-\mu_1,k+c-1,k+c-1,\lambda_1,\ldots,\lambda_{n-2-c}),(-1)^{|\lambda|}.$$

The parametrization is motivated by tile tables for $n=3,4,6$ and by the appearance of conjugate partitions. The source reports it as conjectural and gives no proof.