The finite reduction conjecture for fundamental representations

Let $I_0$ index the fundamental representations, let $\bar k=\bigcup_{n>0}\mathbb C((q^{1/n}))$, and let $\mathfrak m=\bigcup_{n>0}q^{1/n}\mathbb C[[q^{1/n}]]$. For every $i\in I_0$, choose $N\in\mathbb N$, nonzero $b_1,\ldots,b_N,c_1,\ldots,c_N\in\mathfrak m$, indices $s_1,\ldots,s_N,t_1,\ldots,t_N\in I_0$, irreducible finite-dimensional $U'_q({\mathfrak g})_{\bar k}$-modules $W_\mu$, and $U'_q({\mathfrak g})_{\bar k}$-linear maps $\varphi_\mu:V(\varpi_i)\otimes V(\varpi_{s_\mu})_{b_\mu}\to V(\varpi_{t_\mu})_{c_\mu}\otimes W_\mu$. Define $F_0=\bigoplus_{\xi\ne-\varpi_{i^*}}V(\varpi_i)_\xi$ and $F_\mu=\{v\in F_{\mu-1}\mid\varphi_\mu(v\otimes u_{s_\mu})=0\}$. Finite reduction conjecture. These data can be chosen so that $F_N=\bar k u_i$; $\varphi_\mu(F_{\mu-1}\otimes u_{s_\mu})\subset V(\varpi_{t_\mu})_{c_\mu}\otimes w_\mu$; $V(\varpi_{s_\mu})_{b_\mu}$ is not isomorphic to $V(\varpi_{t_\mu})_{c_\mu}$; and $V(\varpi_{s_\mu})_{b_\mu}$ is not a component of $W_\mu$, where $u_{s_\mu}$ and $w_\mu$ are the dominant extremal vectors. This is an auxiliary conjecture used to reduce the paper's main conjecture; its general status is not resolved in the supplied text.