The characterization of totally positive sequence preservers by their remainders

Let ${\mathbf A}=(a_k)_{k=0}^\infty$ be a non-negative sequence whose generating function is entire. For each $l\in\mathbb N\cup\{0\}$, consider the formal power series

$$R_l[\mathbf A](z)=\sum_{k=l}^\infty a_k z^k.$$

The remainder characterization conjecture. The sequence ${\mathbf A}$ is a $\mathrm{TP}_\infty$-preserver if and only if, for every $l\in\mathbb N\cup\{0\}$, $R_l[\mathbf A]$ is an entire function in the $\mathcal{L-P}I$ class; in particular, it has only real nonpositive zeros.

This conjecture seeks to characterize coefficientwise convolution operators preserving totally positive sequences through the zero sets of all their remainders. The preceding theorem establishes the necessity of the $\mathcal{L-P}I$ condition for a $\mathrm{TP}_\infty$-preserver, while the converse remains the conjectural part.