Characterization of totally positive sequence preservers

Let ${\mathbf A}=(a_k)_{k=0}^\infty$ be a nonnegative sequence. For a sequence $(b_k)_{k=0}^\infty$, write

$$\Lambda_{\mathbf A}((b_k)_{k=0}^\infty)=(a_kb_k)_{k=0}^\infty.$$

The sequence ${\mathbf A}$ is a $TP_\infty$-preserver if it sends every sequence in $TP_\infty$ to a sequence in $TP_\infty$.

The $TP_\infty$-preserver conjecture. The sequence ${\mathbf A}$ is a $TP_\infty$-preserver if and only if, for every $l\in\mathbb{N}\cup\{0\}$, the formal power series

$$\sum_{k=l}^\infty a_k z^k$$

is an entire function in the $\mathcal{L-P} I$ class; in particular, it has only real nonpositive zeros.

This would characterize all diagonal preservers of total positivity in terms of the Laguerre–Pólya class. The statement is presented as consistent with the preceding theorems and example, but no resolution is given here.