Fundamental-representation conjecture for quantum cluster variables

For each $(i,r)\in\hat{I}$, let $\tilde{\chi}_{i,r}\in K_t(\mathcal{O}^+_{\mathbb{Z}})$ be the quantum cluster variable obtained from the initial quantum seed by the prescribed mutation sequence, and let $L(Y_{i,q^{r+1}})$ be the fundamental representation with $(q,t)$-character $[L(Y_{i,q^{r+1}})]_t$. Let $\mathcal{J}$ be the morphism into the ambient quantum torus. Fundamental-representation conjecture. For all $(i,r)\in\hat{I}$,

$$\tilde{\chi}_{i,r}=\mathcal{J}\left([L(Y_{i,q^{r+1}})]_t\right).$$

The source verifies this in the displayed $\mathfrak{sl}_2$ example; the assertion for all $(i,r)$ is presented as the resulting conjecture.