Conjecture on the row sum polynomials of the production matrix of an almost Riordan array

Let $P_n(x)$ denote the row sum polynomials of the matrix generated by regarding $\overline{A}$ as a production matrix. The initial polynomials are

$$1,\quad 2-x,\quad x^2-3x+3,\quad -x^3+5x^2-8x+5,\quad x^4-8x^3+22x^2-25x+11,\ldots.$$

Row sum polynomial conjecture. For every $n\geq 0$,

$$P_n(x)=1+(1-x)\sum_{i=0}^{n-1}(1-x)_i.$$

Here $(1-x)_i$ denotes the falling factorial $(1-x)(-x)\cdots(2-x-i)$, with the empty sum interpreted as $0$ when $n=0$. This conjecture gives a closed formula for the row sums of the production matrix and extends the observed initial values; its resolution is not indicated in the supplied text.