Recurrence conjecture for the polynomial factor in the F-determinants

Let $\operatorname{Pol}_n(x)$ be the monic polynomial of degree $2n-2$ occurring in the factorization of the determinant in Theorem MS1. Polynomial-factor recurrence conjecture. The sequence $\operatorname{Pol}_n(x)$ satisfies

$$3\operatorname{Pol}_{n+3}(x)-2(18n^2+9nx+72n-3x^2-3x+49)\operatorname{Pol}_{n+2}(x)$$ $$+(135n^4+108n^3x+810n^3-54n^2x^2+108n^2x+1395n^2-52nx^3-510nx^2-1100nx+120n-9x^4-152x^3-855x^2-1780x-1020)\operatorname{Pol}_{n+1}(x)$$ $$-6(n+1)(n-x-2)(n+x+2)(3n+x+3)(3n+x+5)(3n+x+7)\operatorname{Pol}_n(x)=0,$$

with initial values

$$\operatorname{Pol}_1(x)=1,\qquad \operatorname{Pol}_2(x)=\frac{3x^2+31x+60}{3},\qquad \operatorname{Pol}_3(x)=\frac{9x^4+234x^3+2061x^2+6956x+7680}{9}.$$

The polynomial factor arises in a determinant evaluation for a parameterized family. The source explicitly presents its precise description as an open problem, and no explicit formula for these polynomials is known there.