Let $X$ be a Noetherian normal scheme. Denote by $B$ the singular locus of $X$. By normality, points in $B$ have codimension strictly greater 1. If $X$ is of finite type over field (or some J-2 ring), then it is known that $B$ is closed.

I am interested in the general situation without the J-2/finite type hypothesis. Specifically, I want to know the following:

Let $X$ be a Noetherian normal scheme and $\bar{B}$ denote the closure of the singular locus $B$. Then is codim$(\bar{B})\geq 2$?