Brieskorn (1970) showed that for semiuniversal deformation of rational double points surface singularities $X\to S$, there is a finite base change $S'\to S$, such that the new family $f:X\times_{S}S'\to S'$ admit simultaneous resolution, i.e., there is a a smooth variety $W$ and commutative diagram $\require{AMScd}$ \begin{CD} W @>{\sigma}>> X\times_SS'@>>>X\\ @VV{\sigma}V @VVfV@VVV\\ S' @=S'@>>>S\tag{1}\label{1} \end{CD} ​ where $\sigma$ is a birational morphism which desingularizes each fiber of $f$ and $\pi$ is a smooth morphism. Moreover, Brieskorn showed that the base change is identified with Weyl group action on Cartan subalgebra. (A good reference would be P. Slodowy's Four Lectures on simple groups and singularities.)​ ​

I'd like to know if it is possible to generalize this construction to a proper family.

More precisely, assume $T$ is a smooth projective variety and $Y\to T$ is a flat family of algebraic surfaces with discrimant locus (let's say a divisor) $D\subset T$, and the singularity on each fiber is at worst rational double point. Moreover, suppose for each point $x\in D$, there is a neighborhood $U$ of $x$ in $T$ such that the restricted family $Y_U\to U$ is semiuniversal deformation of ordinary double point singularities (cross some trivial directions).​ ​

Question: Is there a finite base change $T'\to T$ such that $Y\times_TT'\to T'$ admits simultaneous resolution?​ ​

Brieskorn (1970) showed that for semiuniversal deformation of rational double points surface singularities $X\to S$, there is a finite base change $S'\to S$, such that the new family $f:X\times_{S}S'\to S'$ admit simultaneous resolution, i.e., there is a a smooth variety $W$ and commutative diagram $\require{AMScd}$ \begin{CD} W @>{\sigma}>> X\times_SS'@>>>X\\ @VV{\sigma}V @VVfV@VVV\\ S' @=S'@>>>S \end{CD} ​ where $\sigma$ is a birational morphism which desingularizes each fiber of $f$ and $\pi$ is a smooth morphism. Moreover, Brieskorn showed that the base change is identified with Weyl group action on Cartan subalgebra. (A good reference would be P. Slodowy's Four Lectures on simple groups and singularities.)​ ​

For example, semiuniversal deformation of $A_2$ singularity is $$\{x^2+y^2+z^3+tz+s=0\}\to \mathbb C^2, \ (x,y,z,t,s)\mapsto (t,s),\tag{1}\label{1}$$ with the discriminant locus the cuspidal curve $27s^2+4t^3=0$. Now, permutation action on the hyperplane $\{u+v+w=0\}\subset \mathbb C^3$ gives a $6:1$ map $$\{u+v+w=0\}\to \mathbb C^2, \ (u,v,w)\mapsto (uv+uw+vw,uvw).$$ Brieskon's theorem says base change of $(\ref{1})$ along this $6:1$ cover, the total family admits a simultaneous resolution.

I'd like to know if it is possible to generalize this construction to a family with proper base.

More precisely, assume $T$ is a smooth projective variety and $Y\to T$ is a flat family of algebraic surfaces with discrimant locus (let's say a divisor) $D\subset T$, and the singularity on each fiber is at worst rational double point. Moreover, suppose for each point $x\in D$, there is a neighborhood $U$ of $x$ in $T$ such that the restricted family $Y_U\to U$ is semiuniversal deformation of rational double point singularities (cross some trivial directions).​ ​

Question: Is there a finite base change $T'\to T$ such that $Y\times_TT'\to T'$ admits simultaneous resolution?​ ​