Existence of ACM curves realizing a given numerical semigroup of type t

Establish whether for every numerical semigroup T of Cohen–Macaulay type t > 1 there exists an arithmetically Cohen–Macaulay projective monomial curve C of type t whose associated numerical semigroup S_C equals T.

Background

The authors construct arithmetically Gorenstein (type 1) projective monomial curves from symmetric numerical semigroups, showing that for type 1 every symmetric numerical semigroup arises as S_C for some Gorenstein projective monomial curve C.

They pose the broader question of whether this can be extended to higher Cohen–Macaulay types, asking for the existence of arithmetically Cohen–Macaulay curves of type t > 1 that realize a given numerical semigroup T as S_C.