Odd-level graded twisting equivalence conjecture for quantum group representation categories

Let $\mathfrak{g}$ be a simple Lie algebra, let $P$ be its weight lattice and $Q$ its root lattice, and let $Q_{\mathfrak{g}}^N:P/Q\to\mathbb{C}^\times$ be the quadratic form defined by

$$Q_{\mathfrak{g}}^N(\lambda)=\exp\bigl(N\pi i\langle\langle\lambda,\lambda\rangle\rangle\bigr).$$

Let $(U_q(\mathfrak{g}),R(\rho))\text{-mod}^{Q_{\mathfrak{g}}^N}$ denote the graded twist of the braided tensor category $(U_q(\mathfrak{g}),R(\rho))\text{-mod}$ by the abelian cocycle associated with this quadratic form. The graded twisting conjecture. For any $N\in\mathbb{Z}$, the categories

$$(U_q(\mathfrak{g}),R(\rho))\text{-mod}^{Q_{\mathfrak{g}}^N}\quad\text{and}\quad (U_{(-1)^Nq}(\mathfrak{g}),R(\rho+N))\text{-mod}$$

are equivalent as braided tensor categories.

The conjecture is motivated by the established even-shift case, where the analogous equivalence is proved with $Q_{\mathfrak{g}}^{2N}$ and shift $2N$. The supplied excerpt gives no resolution of the stated all-integer conjecture.