Finite-difference convolution conjecture for hyperbolic polynomials

For polynomials of degree at most $d$, define the forward difference operator by

$$\nabla p(x)=p(x+1)-p(x)$$

and define the product

$$(p \bullet q)(x)=\sum_{k=0}^d (\nabla^k p)(0)\,(\nabla^{d-k}q)(x).$$

Here a hyperbolic polynomial has mesh at least one when consecutive roots are separated by at least one. Finite-difference convolution conjecture. If $p$ and $q$ are hyperbolic polynomials of degree at most $d$ and of mesh at least one, then $p\bullet q$ is also hyperbolic and has mesh at least one.

The source presents this as an equivalent, more attractive formulation of the finite-degree difference-operator conjecture. It remains open in the supplied source.