Schreyer's singular-locus conjecture for countable Cohen–Macaulay type

Let $R$ be an analytic Cohen–Macaulay local ${\mathcal C}$-algebra of countable Cohen–Macaulay type. The Schreyer conjecture. $R$ has at most a one-dimensional singular locus, namely,

$$\dim R/\mathfrak p\leq 1$$

for any $\mathfrak p\in\operatorname{Sing} R$. This conjecture was proved by Huneke and Leuschke, who established a stronger statement for excellent Cohen–Macaulay local rings that are complete or have uncountable residue field; the present paper notes that excellence is unnecessary.