Finite-degree hyperbolicity-preserver conjecture for difference operators

Let $T$ be a constant-coefficient difference operator

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

It acts on hyperbolic polynomials of degree at most $m$ whose mesh is at least one, where the mesh is the minimum distance between consecutive roots. Finite-degree difference analogue. The operator $T$ preserves this set if and only if the polynomial $T((x)_m)$ is hyperbolic and has mesh at least one.

This is proposed as a finite-difference analogue of the finite-degree Hermite–Poulain theorem for constant-coefficient differential operators. Its status is open in the supplied source.