  We prove two special cases of a conjecture of J. Fern\'andez de Bobadilla for
hypersurfaces with $1$-dimensional critical loci.
  We do this via a new numerical invariant for such hypersurfaces, called the
beta invariant, first defined and explored by the second author in 2014. The
beta invariant is an algebraically calculable invariant of the local ambient
topological-type of the hypersurface, and the vanishing of the beta invariant
is equivalent to the hypotheses of Bobadilla's conjecture.
