Liu–Wang's Narayana transformation conjecture on log-convexity

Let $N(n,k)=\frac{1}{n}\binom{n}{k}\binom{n}{k+1}$ be the Narayana numbers. For a sequence $(a_k)_{k\geq 0}$ of positive real numbers, define its Narayana transform by

$$b_n=\sum_{k=0}^n N(n,k)a_k,\qquad n\geq 0.$$

A sequence $(a_k)_{k\geq 0}$ is log-convex if $a_{k+1}a_{k-1}\geq a_k^2$ for every $k\geq 1$. Liu–Wang's conjecture. If $(a_k)_{k\geq 0}$ is log-convex, then its Narayana transform $(b_n)_{n\geq 0}$ is also log-convex. The claim is presented as a conjecture of Liu and Wang; the supplied text does not state a resolution.