Wilf's conjecture for numerical semigroups with large second generator

We study Wilf’s conjecture for numerical semigroups $S$ such that the second least generator $a_2$ of $S$ satisfies $a_2>\frac{c(S)+\mu (S)}{3}$, where $c(S)$ is the conductor and $\mu (S)$ the multiplicity of $S$. In particular, we show that for these semigroups Wilf’s conjecture holds when the multiplicity is bounded by a quadratic function of the embedding dimension.