Eulerian transformation preserves log-convexity

Let $A(n,k)$ denote the Eulerian numbers, and let $(x_k)_{k\geq 0}$ be a sequence for which log-convexity is defined. Define its Eulerian transformation by

$$z_n=\sum_{k=0}^{n}A(n,k)x_k.$$

Eulerian transformation conjecture. The Eulerian transformation preserves log-convexity. The source presents this as a related problem after discussing q-log-convexity of Eulerian polynomials; the supplied passage does not specify the input-sequence hypotheses or report a resolution.