Resolution of singularities of the cotangent sheaf

Let $X_0$ be a reduced algebraic variety over a field of characteristic zero, or a reduced complex- or real-analytic variety, and let $n=\dim X_0$. A resolution of singularities is a proper birational or bimeromorphic morphism

$$\sigma:X\to X_0$$

such that $X$ is smooth, $\sigma$ is an isomorphism over $X_0\backslash \operatorname{Sing} X_0$, and $\sigma^{-1}(\operatorname{Sing} X_0)$ is the support of a simple normal crossings divisor $E$ on $X$. In local analytic or étale coordinates $(\mathbf u,\mathbf v)=(u_1,\ldots,u_s,v_1,\ldots,v_{n-s})$ on $X$, differential monomials are forms of the types $d(\mathbf u^{\boldsymbol\alpha_i})$ and $d(\mathbf u^{\boldsymbol\beta_j}v_j)$. Resolution of singularities of the cotangent sheaf. There is a resolution of singularities $\sigma:X\to X_0$ such that the pulled-back cotangent sheaf of $X_0$ is locally generated by

$$d(\mathbf u^{\boldsymbol\alpha_i}),\quad i=1,\ldots,s,\qquad d(\mathbf u^{\boldsymbol\beta_j}v_j),\quad j=1,\ldots,n-s,$$

where

$$\operatorname{supp}E=(u_1\cdots u_s=0),$$

the multiindices $\boldsymbol\alpha_1,\ldots,\boldsymbol\alpha_s\in\mathbb N^s$ are linearly independent over $\mathbb Q$, and $\{\boldsymbol\alpha_i,\boldsymbol\beta_j\}$ is totally ordered with respect to the componentwise partial ordering of $\mathbb N^s$. This conjecture asks for a global resolution making the pulled-back cotangent sheaf monomial in suitable local coordinates; it is proved in dimensions up to three, while the general higher-dimensional case remains open.