Jacobi-Stirling transformation conjecture on preserving log-convexity

A sequence transformation maps a sequence $\{z_n\}_n$ to a sequence $\{w_n\}_n$. For the Jacobi-Stirling numbers of the second kind $\operatorname{JS}(n,k;z)$ and the Jacobi-Stirling numbers of the first kind $\operatorname{Jc}(n,k;z)$, define

$$w_n=\sum_{k=0}^n\operatorname{JS}(n,k;z)z_k$$

or

$$w_n=\sum_{k=0}^n\operatorname{Jc}(n,k;z)z_k.$$

A sequence is log-convex if $z_n^2\leq z_{n-1}z_{n+1}$ for the relevant indices. Jacobi-Stirling transformation conjecture. Each of these transformations preserves log-convexity for $z=0$ and $z=1$: whenever $\{z_n\}_{n\geq0}$ is log-convex, the resulting sequence $\{w_n\}_{n\geq0}$ is also log-convex. This extends the known log-convexity-preservation result for the Stirling transformation and is posed in connection with the log-convexity properties of Jacobi-Stirling polynomials; its resolution is not given in the source.