Liu–Wang's log-convexity conjecture for the Eulerian transformation

Let $\{x_n\}_{n\geq 0}$ be a sequence of nonnegative numbers, and define $z_n=\sum_{k=0}^n A_{n,k}x_k$ using the Eulerian triangle $\{A_{n,k}\}_{0\leq k\leq n}$. A sequence $\{a_n\}_{n\geq 0}$ is log-convex if $a_k^2\leq a_{k-1}a_{k+1}$ for every $k\geq 1$. Liu–Wang's conjecture. The Eulerian transformation preserves log-convexity: whenever $\{x_n\}$ is log-convex, so is $\{z_n\}$. Liu and Wang proposed this conjecture after establishing analogous results for the binomial and Stirling transformations. It is stated in the source as still open.