Narayana transformation preserves log-convexity

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

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

Narayana transformation conjecture. The Narayana transformation preserves log-convexity. The source presents this as a problem following known log-convexity results for specializations of the Narayana polynomials; the required hypotheses on the input sequence are not made explicit in the supplied passage.