The automorphism-group conjecture for quantum matrices

Let $\mathbb{K}$ be a field, let $q\in\mathbb{K}^*$ be not a root of unity, and let ${\mathcal O}_q(M_n)$ be the quantisation of the ring of regular functions on $n\times n$ matrices. Let $\mathcal{H}$ be the subgroup of automorphisms acting on each matrix generator $Y_{i,\alpha}$ by multiplication by a nonzero scalar, and let $\tau$ be the transposition automorphism sending $Y_{i,\alpha}$ to $Y_{\alpha,i}$. Automorphism-group conjecture.

$${\rm Aut}({\mathcal O}_q(M_n))=\mathcal{H}\rtimes\langle\tau\rangle.$$

This predicts that the known scalar automorphisms and, in the square case, transposition generate the full automorphism group of quantum matrices. The corresponding result is known for $n=2$, while the preceding discussion explains that the methods available for nonsquare quantum matrices do not resolve the general square case.