Maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum

We say that a Cohen-Macaulay local ring has finite $\operatorname{\mathsf{CM}}_+$-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. In this paper, we consider finite $\operatorname{\mathsf{CM}}_+$-representation type from various points of view, relating it with several conjectures on finite/countable Cohen-Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite $\operatorname{\mathsf{CM}}_+$-representation type are exactly the local hypersurfaces of countable $\mathsf{CM}$-representation type, that is, the hypersurfaces of type $(\mathrm{A}_\infty)$ and $(\mathrm{D}_\infty)$. We also discuss the closedness and dimension of the singular locus of a Cohen-Macaulay local ring of finite $\operatorname{\mathsf{CM}}_+$-representation type.