Hypergeometric closed form for the excursions of the second Kreweras-like model

Consider the model

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

Let $Q(0,0;t^{1/2})$ be its excursion generating function after extracting the even subsequence. The hypergeometric closed-form conjecture.

$$Q(0,0;t^{1/2})=\frac{1}{3t}-\frac{\sqrt{1-12t}}{6t}\left({}_2F_1\left(\frac16,\frac13;1;\frac{-108t(1+4t)^2}{(1-12t)^3}\right)+{}_2F_1\left(-\frac16,\frac23;1;\frac{-108t(1+4t)^2}{(1-12t)^3}\right)\right).$$

The formula is inferred from the guessed differential operator and its hypergeometric solutions, so it remains conjectural rather than proved in the source.