Tokuyama-type character formula for $G_2$

Let $\Phi=G_2$. For a dominant weight $\theta$, let $\chi_\theta$ be the character of the irreducible representation $V_\theta$ of $G_2$ of lowest weight $-\theta$, shifted to be an element of $\mathbb{C}[\Lambda]$. Define the deformed Weyl denominator

$$D(\mathbf{x})=\prod_{\alpha>0}(1-q^{-1}\mathbf{x}^{\alpha})$$

and set $N_\theta(\mathbf{x})=\chi_\theta(\mathbf{x})D(\mathbf{x})$. Let $\mathscr{B}(\theta+\rho)$ denote the set of $G_2$-patterns of highest weight $\theta+\rho$, and let $\widehat{H}(\pi)$ be the modified contribution associated with such a pattern. Tokuyama-type conjecture for $G_2$. One has

$$N_\theta(\mathbf{x})=\sum_{\pi\in\mathscr{B}(\theta+\rho)}\widehat{H}(\pi)\mathbf{x}^{\pi}.$$

This conjecture proposes a crystal-graph expansion of the deformed Weyl character numerator for $G_2$, extending Tokuyama's theorem beyond the classical settings where such formulas are established. The supplied text does not state whether the conjecture has been proved or disproved.