Conjecture on Robbins numbers from an amalgamation of Riordan arrays

For two Riordan arrays $A=(a_{n,k})$ and $B=(b_{n,k})$, define their amalgamation $A\amalg B=(t_{n,k})$ by

$$t_{n,k}=\begin{cases}a_{n,n-k}&\text{if }k\leq n,\\ b_{k,k-n}&\text{otherwise}.\end{cases}$$

Consider

$$\left(\frac{1-x}{1-x+x^2},\frac{x}{1-x}\right)\amalg\left(\frac{1-x}{1-x+x^2},\frac{-x}{1-x}\right).$$

Robbins-number amalgamation conjecture. The principal minor sequence of this amalgamation is the sequence of Robbins numbers.

This is a second formulation of the preceding Robbins-number claim, using Riordan arrays and principal minors; the source supplies no resolution.