The higher production matrix formula for Riordan arrays

Let $M=(g(x),f(x))$ be a Riordan array, and let $P^{(n)}_M$ denote its $n$-th production matrix, defined by

$$P^{(n)}_M=|^{(n-1)}M^{-1}\overline{M}^{(n)}.$$

Let $M^{(n)}$ be the matrix produced by $P^{(n)}_M$. Higher production matrix conjecture. For every positive integer $n$,

$$M^{(n)}=\left(\left(\frac{x}{f(x)}\right)^{n-1},x\left(\frac{x}{f(x)}\right)^{n-1}\right)^{-1}\cdot(g(x),f(x)).$$

This generalizes the preceding formulas for the second and third production matrices of a Riordan array. The source presents the formula as a conjecture, and no resolution is supplied in the cited text.