Type-monotonicity conjecture for numerical semigroups

Let $n(g,t)$ denote the number of numerical semigroups of genus $g$ and type $t$. Type-monotonicity conjecture. For each pair of integers satisfying $2\leqslant t\leqslant g$,

$$n(g,t)<n(g+1,t).$$

This is proposed as a type-refined analogue of the still-open conjecture that $n(g)<n(g+1)$ for every $g\geqslant 1$. Its general status is open.