Consider a commutative group $G$ of finite type, a subgroup of finite index $H\subseteq G$, a noetherian commutative ring $A$, and a $G$-graded $A$-algebra $R=\bigoplus_{g\in G}R_g$ with no zero-divisors, and denote by $R_H=\bigoplus_{g\in H}R_g$ the degree restriction of $R$ to $H$.

It is well-known that if $R$ is of finite type over $A$, then so is $R_H$. So, we can ask whether the following statement is true:

(+) If $R_H$ is of finite type over $A$, then so is $R$.

Using the fact that $R$ is integral over $R_H$, one can show that (+) holds in the following two cases:

$R_H$ is integrally closed and the field of fractions of $R$ is a separable extension of the field of fractions of $R_H$;

$R_H$ is a japanese ring (see EGA 0$_{\rm IV}$.23).

In particular, (+) holds if $A$ is universally japanese, e.g. excellent, e.g. of finite type over a field.

My question is now as follows:

Is there an example where (+) does not hold?