  In a previous paper, the first three authors formulated a precise conjecture
about the dimension of the {\it generalized Severi variety} $M^n_{d,g; {\rm S},
{\bf k}}$ of degree-$d$ holomorphic maps $\mathbb{P}^1 \rightarrow
\mathbb{P}^n$ whose images' singularities are singleton cusps with value
semigroups ${\rm S}$ and ramification profiles ${\bf k}$. In this paper, we
prove that an adjusted form of the conjecture holds for generic profiles ${\bf
k}$ associated with two distinguished (infinite) classes of semigroups ${\rm
S}$.
