Pantone's algebraicity conjecture for inversion sequences avoiding 010 and 102

Let $\mathcal{I}(010,102)$ be the class of inversion sequences avoiding the patterns $010$ and $102$, and let $F(x)$ be its ordinary generating function. Pantone's algebraicity conjecture. The generating function $F$ is algebraic, with minimal polynomial

$$x(x^2-x+1)(x-1)^2F(x)^3+2x(x-1)(2x^2-2x+1)F(x)^2-(x^4-8x^3+11x^2-6x+1)F(x)-(2x-1)(x-1)^2.$$

The paper presents this as another conjecture of Pantone; the supplied text gives no resolution, so its status remains open.