The versal torsor conjecture for reductive groups over algebraically closed fields

Let $k$ be an algebraically closed field, let $G$ be a connected reductive linear algebraic group over $k$, let $R$ be a regular local ring containing $k$, and let

$$\tau\colon T\to\operatorname{Spec}(R)$$

be a $G$-torsor. Let $K$ be the field of fractions of $R$, and let

$$\tau_K\colon T_K\to\operatorname{Spec}(K)$$

be the $G$-torsor obtained by restricting $\tau$ to the generic point of $\operatorname{Spec}(R)$. Assume that the essential dimension of $\tau_K$ over $k$ is $d$, written $\operatorname{ed}_k(\tau_K)=d$. Versal torsor conjecture. There exists a Cartesian diagram of $k$-morphisms

\xymatrix{ T \ar@{->}[d]_{\tau} \ar@{->}[rr] & & W \ar@{->}[d]^{\nu} \\ \operatorname{Spec}(R) \ar@{->}[rr] & & Y, }

where $Y$ is a $d$-dimensional integral scheme of finite type over $k$ and $\nu$ is a $G$-torsor.

The claim is a proposed variant of the Grothendieck-Serre conjecture, relating the essential dimension of the generic torsor to a finite-type geometric model over the base field. The supplied source gives no resolution or partial status for this proposed statement.