Total-positivity conjecture for the reversed Stirling subset triangle

Let \genfrac\{\}{\}{0pt}{}{n}{k} denote the Stirling subset number, and define the reversed Stirling subset triangle by \genfrac\{\}{\}{0pt}{}{n}{k}^{\rm rev}=\genfrac\{\}{\}{0pt}{}{n}{n-k}. Write \bm{S}^{\rm rev}=\left(\genfrac\{\}{\}{0pt}{}{n}{k}^{\rm rev}\right)_{n,k\geq 0}. Total-positivity conjecture. The reversed Stirling subset triangle $\bm{S}^{\rm rev}$ is totally positive. The source emphasizes that total positivity need not be preserved by reversal and presents this assertion as an open conjecture; it later notes that the more general conjecture would imply it by specialization.