Sabbah's conjecture on good formal structures for surface connections

Let $X$ be a smooth complex surface, let $D$ be a normal-crossings divisor, let $x_0$ be a point of $D$, and let ${\cal M}$ be a meromorphic connection on $X$ with poles along $D$. A connection has a good formal structure if, after a bicyclic ramification along the components of the divisor, its inverse image has a good formal decomposition into elementary models.

Sabbah's conjecture. There is a finite sequence of point blow-ups

$$b:X'\to X$$

over $x_0$ such that the inverse image connection $b^*{\cal M}$ has a good formal structure in a neighborhood of $x_0$.

This is the surface analogue of the Levelt–Turrittin formal decomposition theorem for meromorphic connections on curves. The statement concerns resolving the obstruction to a good formal structure by blowing up points on the polar divisor; the supplied source does not indicate whether the conjecture is resolved.