The ordered tensor-product generation and submodule conjecture for fundamental representations

Let $I_0$ index the fundamental representations, let $i_1,\ldots,i_l\in I_0$, and let $a_1,\ldots,a_l\in\bar k^\times$, where $\bar k=\bigcup_{n>0}\mathbb C((q^{1/n}))$. Write $a\leq b$ when $a/b\in\bar A$, with $\bar A=\bigcup_{n>0}\mathbb C[[q^{1/n}]]$, and let $u_i$ be the dominant extremal vector of $V(\varpi_i)$. The ordered tensor-product conjecture. If $a_1\leq\cdots\leq a_l$, then $V(\varpi_{i_1})_{a_1}\otimes\cdots\otimes V(\varpi_{i_l})_{a_l}$ is generated by $u_{i_1}\otimes\cdots\otimes u_{i_l}$ as a $U'_q({\mathfrak g})_{\bar k}$-module; if $a_1\geq\cdots\geq a_l$, then every nonzero $U'_q({\mathfrak g})_{\bar k}$-submodule contains this tensor. This conjecture concerns the generation and irreducibility behavior of ordered tensor products of fundamental representations; the paper proves it for types $A_n^{(1)}$ and $C_n^{(1)}$, while it is not established in the generality stated here.