Let $f: (X,x)\rightarrow (\mathbb C,0)$ be a deformation of a curve singularity $(X_0,x)$, and let $f: X \rightarrow T$ be a sufficiently small representative. Assume that $(X,x)$ is reduced and pure dimensional, and for all $t\in T$, $t\not =0$, the fibers $X_t:=f^{-1}(t)$ are smooth curve singularities. Is it true then that $X_0$ is reduced at $x$?

I have a counter example for this question in the case $X$ is reduced but not pure dimensional (for that example, $X_t$ are smooth for all $t\not =0$, but they have two isolated points, whereas $X_0$ has an embedded non-reduced point at $0$).