I would like to know a reference of the following statement (or counter example).

Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is Noetherian local and $S_1$ is finitely generated over $S_0$. If $S$ is an integral domain, then $X = \operatorname{Proj} S$ is a irreducible and reduced. Hence $\mathcal{O}_{X,x}$ is an integral domain for any $x \in X$. But what about equidimensionality? That is

Let $S$ be a Noetherian standard graded equidimensional ring over a Noetherian local ring. Then is the local ring $\mathcal{O}_{X,x}$ equidimensional for all $x \in X$?

A ring $R$ is called $\textit{equidimensional}$ if $\dim R = \dim R/p$ for any minimal prime $p$ of $R$.