Coefficientwise total positivity of some matrices defined by linear recurrences

We exhibit a lower-triangular matrix of polynomials $T(a,c,d,e,f,g)$ in six indeterminates that appears empirically to be coefficientwise totally positive, and which includes as a special case the Eulerian triangle. We prove the coefficientwise total positivity of $T(a,c,0,e,0,0)$, which includes the reversed Stirling subset triangle.
Comment: LaTeX2e, 12 pages with 2 figures: extended abstract submitted to FPSAC 2021