Betti sequences of four-generated non-almost-symmetric nearly Gorenstein numerical semigroup rings

Let \(S\) be a minimally four-generated numerical semigroup that is nearly Gorenstein but not almost symmetric, and let \(R=\Bbbk[S]\) over an arbitrary field \(\Bbbk\). We prove that the defining ideal of \(R\) has four or five minimal generators when \(R\) has Cohen--Macaulay type two, and exactly six when it has type three. Thus the total Betti sequence over the four-variable polynomial presentation ring is one of\[(1,4,5,2),\qquad (1,5,6,2),\qquad (1,6,8,3).\]This three-sequence classification gives an affirmative answer, over every field, to Moscariello and Strazzanti's Question 4.3 on four-generated nearly Gorenstein numerical semigroups that are not almost symmetric.