The $\mathbf{i}$-trails of Berenstein and Zelevinsky was introduced on page 5 (Definition 2.1) in this paper. It is defined as follows. Let $\gamma, \delta \in \mathfrak{h}^*$. Let ${\bf i}=(i_1, \ldots, i_l)$. Then $\pi=(\gamma = \gamma_0, \gamma_1, \ldots, \gamma_l=\delta)$ is called an ${\bf i}$-trail if for $k=1,\ldots, l$, $\gamma_{k-1}-\gamma_{k} = c_k\alpha_{i_k}$ for some $c_k \in \mathbb{Z}_{\geq 0}$ and $e_{i_1}^{c_1} \cdots e_{i_l}^{c_l}$ is a non-zero linear map from $V(\delta)$ to $V(\gamma)$.

One main result in the paper is a polyhedral formula for tensor product multiplicities (Theorems 2.2 and 2.3 in the above paper).

My question is: let $\mathfrak{g}$ be the simple complex Lie algebra of type $G_2$. Let $\lambda=k_1 \omega_1 + l_1 \omega_2$ and $\mu=k_2 \omega_1 + l_2 \omega_2$, where $\omega_1, \omega_2$ are fundamental weights. Are there some polyhedral formula which describe the decomposition of $V(\lambda) \otimes V(\mu)$ into irreducible $\mathfrak{g}$-modules using only $k_1, k_2, l_1, l_2$? The formula will be of the form $$ V(\lambda) \otimes V(\mu) = \oplus_{(r_1, r_2) \in A} V(r_1 \omega_1 + r_2 \omega_2), \quad (1) $$ where $$A = \left\{(r_1, r_2) \in \mathbb{Z}_{\geq 0}^2 \middle\vert \begin{aligned} r_1 &=g_1(k_1,k_2,l_1,l_2),\\ r_2 &=g_1(k_1,k_2,l_1,l_2),\\ 0\ &\le f_1(k_1,k_2,l_1,l_2),\\ &\quad\ldots,\\ 0\ &\le f_m(k_1,k_2,l_1,l_2) \end{aligned} \right\},$$ and where $f_1,\ldots, f_m,g_1,g_2$ are some (linear) polynomials. Maybe use the definition of ${\bf i}$-trail, we can translate Theorem 2.2 in the paper to the form $(1)$. But I don't know how to verify that $e_{i_1}^{c_1} e_{i_2}^{c_2}$ is a non-zero linear map from $V(s_i \omega_i^{\vee})$ to $V(w_0 \omega_i^{\vee})$ (in the case of type $G_2$, $w_0 = s_1s_2s_1s_2s_1s_2$). Are there some references which have some formula similar to $(1)$? Thank you very much.