Turán transforms preserve eventually increasing shifted multiplier sequences

Let $\{\gamma_i\}_{i\geq 0}$ be a shifted multiplier sequence that is eventually increasing, and define

$$\mathcal{T}_j(\{\gamma_i\}_{i\geq 0})=\{T_j(i)\}_{i\geq 0}.$$

The Turán-transform preservation conjecture. For every $j\in\mathbb{N}$, the sequence $\mathcal{T}_j(\{\gamma_i\}_{i\geq 0})$ is again a shifted multiplier sequence that is eventually increasing. This ambitious statement would imply eventual positivity for arbitrary compositions of Turán inequalities; the source presents it as open.