Premet's conjugacy conjecture for Cartan subalgebras of restricted Lie algebras

Let ${\mathfrak g}$ be a finite-dimensional restricted Lie algebra, and let $G$ be the group acting on ${\mathfrak g}$ by conjugation. For $u\in{\mathfrak g}$, write ${\mathfrak g}^0(\operatorname{ad}u)$ for the associated Cartan subalgebra, and call an element of ${\mathfrak g}$ regular when it is regular in the sense used for restricted Lie algebras.

Premet's conjecture. There exists a nonempty Zariski open subset $V\subseteq{\mathfrak g}$ consisting of regular elements such that, for any $u,v\in V$, the Cartan subalgebras

$${\mathfrak g}^0(\operatorname{ad}u)\quad\text{and}\quad {\mathfrak g}^0(\operatorname{ad}v)$$

are conjugate under $G$.

The conjecture predicts that the Cartan subalgebra attached to a generic regular element is unique up to conjugacy. The supplied passage attributes it to Premet and does not state whether it has been resolved.