  Let $f : X\to \Delta$ be a $1$-parameter family of $2$-dimensional isolated
hypersurface singularities. In this paper, we show that if the Milnor number is
constant, then any semistable model, obtained from $f$ after a sufficiently
large base change must satisfy non trivial restrictions. Those restrictions are
in terms of the dual complex, Hodge structure and numerical invariants of the
central fibre.
