Brändén–Krasikov–Shapiro mesh-preserving difference-operator conjecture

Let

$$T(p(x))=a_0p(x)+a_1p(x-1)+\cdots+a_kp(x-k)$$

be a difference operator with constant coefficients. The mesh of a polynomial with all real simple zeros is the minimal distance between consecutive roots.

Brändén–Krasikov–Shapiro conjecture. The operator $T$ preserves the set of real-rooted polynomials of degree at most $m$ whose mesh is at least $1$ if and only if $T((x)_m)$ is real-rooted and has mesh at least $1$, where

$$(x)_m=x(x-1)(x-2)\dots(x-m+1).$$

The source also gives an equivalent formulation using the forward difference operator, but does not report a resolution of the conjecture.