Monoidality of the cyclic functor from shifted category O to KLR modules

Let $\mathfrak{g}$ be the Lie-theoretic datum of the paper, let $\mathcal O_{\mathrm{sh}}$ be the shifted category O, and let $R=\bigoplus_{\nu\in Q_+}R_\nu$ be the associated KLR algebra. Equip $\bigoplus_{\nu\in Q_+}R_\nu\text{-}\operatorname{fmod}$ with its convolution monoidal structure. Monoidality conjecture for $\Theta_{\operatorname{cyc}}$. The functor

$$\Theta_{\operatorname{cyc}}:\mathcal O_{\mathrm{sh}}\to\bigoplus_{\nu\in Q_+}R_\nu\text{-}\operatorname{fmod}$$

is monoidal. The preceding multiplicativity result gives evidence for monoidality, and the source says that the claim will hold on the level of $K_0$ in forthcoming work; the categorical assertion remains open there.