Parametric algebraic formula for the excursions of a Gessel-like model

Consider the model

$$\mathcal S=\{\bar 2\bar 1,\bar 1 1,0\bar 1,11\}.$$

Let $Q(0,0;t)$ be its excursion generating function. Define $U$ as the unique power series in $\mathbb Q[[t]]$ with

$$U=t^4+53t^8+4363t^{12}+\cdots$$

satisfying

$$U(1-2U)^3(1-3U)^3(1-6U)^9=t^4(1-4U)^4.$$

The parametric algebraic-form conjecture.

$$Q(0,0;t)=\frac{(1-4U)(1-24U+120U^2-144U^3)}{(1-3U)(1-2U)^{3/2}(1-6U)^{9/2}}.$$

This formula is suggested by a guessed algebraic equation and a rational parametrization; the source presents it as conjectural.