Threshold conjecture for third-order Turán inequalities and invariant positivity

Let $p(n)$ and $\bar{p}(n)$ denote the partition functions considered in the paper, let $\Delta^k$ denote the $k$-th finite-difference operator, and let $I$ denote the invariant used in the paper. For $1\leq k\leq 5$, write $T_p(k)$, $T_{\bar{p}}(k)$, $I_p(k)$, and $I_{\bar{p}}(k)$ for the proposed thresholds.

Threshold conjecture. For $1\leq k\leq 5$, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies the third-order Turán inequality whenever $n\geq T_p(k)$, respectively $n\geq T_{\bar{p}}(k)$. Moreover, $\Delta^k p(n)$, respectively $\Delta^k\bar{p}(n)$, satisfies $I>0$ whenever $n\geq I_p(k)$, respectively $n\geq I_{\bar{p}}(k)$.

These conjectures give explicit threshold ranges for two further positivity properties of finite differences of the partition functions. The source specifies the proposed thresholds in tables, while the validity of the assertions remains open.