The bi-Galois conjecture for the twist between Manin's universal quantum groups

Let $V$ be a $4$-dimensional vector space over $\Bbbk$ with basis $\{x_0,x_1,x_2,x_3\}$. Let $H=\underline{\operatorname{aut}}^l(\Bbbk[x_0,x_1,x_2,x_3])$ and $K=\underline{\operatorname{aut}}^l(S(\alpha,\beta,\gamma))$, where $S(\alpha,\beta,\gamma)$ is a $4$-dimensional Sklyanin algebra in the stated nonexceptional parameter range. Let $T$ be the $K$-$H$-bicomodule algebra described by the generators $\mathbb A=(a_{ij})_{1\leq i,j\leq 4}$, $\mathbb B=(b_{ij})_{1\leq i,j\leq 4}$, and $D^{\pm1}$, subject to the displayed relations. A bicomodule algebra is cleft bi-Galois if it is a bi-Galois object admitting cleft structures on both sides. The bi-Galois conjecture. The bicomodule algebra $T$ is nonzero and cleft bi-Galois, and consequently yields a $2$-cocycle twisting $H$ to $K$. This would provide the conjectural explicit realization of the twist relating the universal quantum groups of the polynomial ring and the Sklyanin algebra; the supplied text gives no resolution, so the claim remains open.