Bekka's $(c)$-regularity condition and families of line singularities with constant Lê numbers

We show that the natural stratifications arising from certain deformation families of line singularities with constant Lê numbers satisfy Bekka's $(c)$-regularity condition. As a corollary, we obtain that these families are topologically equisingular. Similar results for families of isolated singularities were established by Abderrahmane.