  This article investigates the monodromy conjecture for a space monomial curve
that appears as the special fiber of an equisingular family of curves with a
plane branch as generic fiber. Roughly speaking, the monodromy conjecture
states that every pole of the motivic, or related, Igusa zeta function induces
an eigenvalue of monodromy. As the poles of the motivic zeta function
associated with such a space monomial curve have been determined in earlier
work, it remains to study the eigenvalues of monodromy. After reducing the
problem to the curve seen as a Cartier divisor on a generic embedding surface,
we construct an embedded $\mathbb Q$-resolution of this pair and use an A'Campo
formula in terms of this resolution to compute the zeta function of monodromy.
Combining all results, we prove the monodromy conjecture for this class of
monomial curves.
