  In 1982, Tamaki Yano proposed a conjecture predicting how is the set of
$b$-exponents of an irreducible plane curve singularity germ which is generic
in its equisingularity class. In 1986, Pi.~Cassou-Nogu\`es proved the
conjecture for the one Puiseux pair case. In a previous work the authors proved
the conjecture for two Puiseux pairs germs whose complex algebraic monodromy
has distinct eigenvalues. A natural problem induced by Yano's conjecture is,
for a generic equisingular deformation of an isolated plane curve singularity
germ to study how the set of $b$-exponents depends on the topology of the
singularity. The natural generalization suggested by Yano's approach holds in
suitable examples (for the case of isolated singularites which are Newton
non-degenerated, commode and whose set of spectral numbers are all distincts).
Morevover we show with an example that this natural generalization is not
correct. We restrict to germs whose complex algebraic monodromy has distinct
eigenvalues such that the embedded resolution graph has vertices of valency at
most 3 and we discuss some examples with multiple eigenvalues.
