Projective closure of Gorenstein monomial curves and the Cohen-Macaulay property

Let $C(\mathbf{a})$ be a Gorenstein non-complete intersection monomial curve in the 4-dimensional affine space. There is a vector $\mathbf{v}\in {\mathbb{N}}^4$ such that for every integer $m\ge 0$, the monomial curve $C(\mathbf{a}+m\mathbf{v})$ is Gorenstein non-complete intersection whenever the entries of $\mathbf{a}+m\mathbf{v}$ are relatively prime. In this paper, we study the arithmetically Cohen-Macaulay property of the projective closure of $C(\mathbf{a}+m\mathbf{v})$.