Double affine braid group action conjecture for 2-representations

Let $\widehat{\widehat{\mathfrak{g}}}$ be the double-affine algebra in the sense of the paper, and let $B_{\widehat{\widehat{\mathfrak{g}}}}$ be its double affine braid group. A double affine braid group action conjecture asserts that every 2-representation of $\widehat{\widehat{\mathfrak{g}}}$ in the sense of the paper induces an action of $B_{\widehat{\widehat{\mathfrak{g}}}}$. This generalizes the established relationship between geometric or strong categorical Kac–Moody actions and braid-group actions; the double-affine case is stated as an expectation in the source.