Threshold conjecture for Laguerre inequalities and determinant positivity of partition-function differences

Let $p(n)$ and $\bar{p}(n)$ denote the partition functions considered in the paper, and let $\Delta^k$ denote their $k$-th finite differences. For $1\leq m\leq 11$ and $1\leq k\leq 5$, write $L_p(k,m)$, $L_{\bar{p}}(k,m)$, $D_p(k,m)$, and $D_{\bar{p}}(k,m)$ for the proposed threshold functions.

Threshold conjecture. For $1\leq m\leq 11$ and $1\leq k\leq 5$, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies the Laguerre inequality of order $m$ whenever $n\geq L_p(k,m)$, respectively $n\geq L_{\bar{p}}(k,m)$. The $m$-th order determinants of $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, are positive whenever $n\geq D_p(k,m)$, respectively $n\geq D_{\bar{p}}(k,m)$, except when both $k\geq 3$ and $m$ are odd.

These assertions propose explicit thresholds for higher-order inequalities of finite differences of the partition functions. The source presents the thresholds in tables; their general validity is conjectural, and the exceptional parity range is part of the stated claim.