Type bound for almost symmetric generalized numerical semigroups

Let $S\subseteq\mathbb{N}^d$ be an almost symmetric generalized numerical semigroup of embedding dimension $2d+2$.

Type-bound conjecture. The type of $S$ can be at most three.

The computational data in the paper support this bound for the parameters considered, and the corresponding statement is known for numerical semigroups, that is, when $d=1$. The conjecture concerns whether the type is universally bounded by three in embedding dimension $2d+2$.