The trigonometric shifted quantum affine injectivity conjecture

Let $\Lambda$ be the coweight lattice, let $\Lambda^+\subset\Lambda$ be the dominant coweights, and let $\Psi_D$ be the homomorphisms defined in~ for $\Lambda$-valued divisors $D$ on $\mathbb{P}^1$. Consider divisors that are $\Lambda^+$-valued outside $\{0,\infty\}\subset\mathbb{P}^1$, satisfy~, and have prescribed restrictions $D|_\infty=\mu^+$ and $D|_0=\mu^-$. Trigonometric kernel-intersection conjecture. For any coweights $\mu^+,\mu^-\in\Lambda$, the intersection of the kernels of these homomorphisms is zero:

$$\bigcap_D \operatorname{Ker}(\Psi_D)=0.$$

Here $D$ ranges over all divisors with the stated properties. This conjecture is the trigonometric counterpart of the cited theorem used to establish injectivity of the map $\Upsilon_{-\mu^+,-\mu^-}$; the supplied text does not state whether it has been proved or remains open.