Eventual positivity of iterated Turán inequalities for shifted multiplier sequences

Let $\{\gamma_i\}_{i\geq 0}$ be a shifted multiplier sequence that is eventually increasing. For $j\in\mathbb{N}$, let $T_j$ denote the order-$j$ Turán operator, and write $T_j^{(k)}$ for its $k$-fold iterate. The iterated Turán-positivity conjecture. For each $j,k\in\mathbb{N}$, there exists $N_j^{(k)}$ such that

$$T_j^{(k)}(n)>0$$

for all $n\geq N_j^{(k)}$. This generalizes eventual infinite log-concavity to all orders of Turán inequalities; the paper gives evidence from partition numbers and discusses several known special cases, but the general assertion remains open.