Yet another criterion for the total positivity of Riordan arrays

Let $R=\mathcal{R}(d(t),h(t))$ be a Riordan array, where $d(t)=\sum_{n\ge 0}d_nt^n$ and $h(t)=\sum_{n\ge 0}h_nt^n$. We show that if the matrix \begin{equation*} \left[\begin{array}{ccccc} d_0 & h_0 & 0 & 0 &\cdots\\ d_1 & h_1 & h_0 & 0 &\\ d_2 & h_2 & h_1 & h_0 &\\ \vdots&\vdots&&&\ddots \end{array}\right] \end{equation*} is totally positive, then so is the Riordan array $R$.