Lin–Zeng's Jacobi–Stirling transformations conjecture

Let $\text{JS}_n^k(z)$ and $\text{Jc}_n^k(z)$ denote the Jacobi–Stirling numbers of the second and first kinds, respectively, and let $(x_k)_{k\geq 0}$ be a sequence. Define the two transformations

$$y_n=\sum_{k=0}^{n}\text{JS}_n^k(z)x_k\quad\text{and}\quad w_n=\sum_{k=0}^{n}\text{Jc}_n^k(z)x_k.$$

Lin–Zeng's conjecture. The Jacobi–Stirling transformations of the two kinds preserve log-convexity for $z=0,1$. This conjecture concerns preservation of log-convexity by the transformations associated with the Jacobi–Stirling numbers. The source attributes it to Lin and Zeng; the supplied material gives no evidence that it has been resolved.