Minkowski-sum lattice-point formula for $G_2$ representations

Let $\lambda=m_1\varpi_1+m_2\varpi_2\in\mathcal{P}_+$, and let $\boldsymbol{G_2^{\varpi_1}}(1)$ and $\boldsymbol{G_2^{\varpi_2}}(1)$ be the polytopes defined by the displayed inequalities in the source. Let $e_3,e_5$ be the corresponding standard basis vectors of $\mathbb{R}^6$. Minkowski-sum formula. The number of lattice points in

$$m_1\boldsymbol{G_2^{\varpi_1}}(1)+m_2\left(\boldsymbol{G_2^{\varpi_2}}(1)\cup\{3e_3,3e_5\}\right)$$

coincides with $\dim V(m_1\varpi_1+m_2\varpi_2)$.

This gives a proposed lattice-point model for dimensions of all finite-dimensional $G_2$ representations, extending the explicitly analyzed fundamental representations. The excerpt does not state whether the formula has been proved or remains open.