Pantone's algebraicity conjecture for inversion sequences avoiding 000 and 102

Let $\mathcal{I}(000,102)$ be the class of inversion sequences avoiding the patterns $000$ and $102$, and let $F(x)$ be its ordinary generating function. Pantone's algebraicity conjecture. The generating function $F$ is algebraic, with minimal polynomial

$$x^4 F(x)^4-2x^3(x-1)F(x)^3+x(x^3-2x^2+4x-1)F(x)^2-(2x^2-2x+1)F(x)+1.$$

The conjecture was formulated from computed initial terms using Pantone's software; the paper reports that Pantone subsequently proved it by calculating the generating function.