  In 1982, Yano proposed a conjecture predicting the $b$-exponents of an
irreducible plane curve singularity which is generic in its equisingularity
class. In this article we prove the conjecture for the case of two Puiseux
pairs and monodromy with distinct eigenvalues. The hypothesis on the monodromy
implies that the $b$-exponents coincide with the opposite of the roots of the
Bernstein polynomial, and we compute the roots of the Bernstein polynomial.
