Characterization of non-trivial discrete multiplier sequences

Let $\{\alpha_i\}_{i=0}^\infty$ be a non-trivial sequence, meaning it is not zero for all but at most two indices, and suppose that $\alpha_i>0$ for some $i$. A discrete multiplier sequence is a sequence whose diagonal finite difference operator preserves $\mathcal{HP}^+_{\ge 1}$. Characterization conjecture. The sequence $\{\alpha_i\}_{i=0}^\infty$ is a discrete multiplier sequence if and only if it is a classical multiplier sequence and

$$0 \leq \alpha_1 \leq \alpha_2 \leq \cdots.$$

This is proposed as a discrete analogue of the classical Pólya–Schur characterization of multiplier sequences. The claim concerns non-trivial sequences and remains open in the supplied source.