A theorem of Knop states that if $G$ is semisimple and connected acting on a vector space $V$ over a field $K$ of characteristic 0, then the degree of the Hilbert series of $K[V]^G$ is less than or equal to $-\operatorname{dim}( K[V]^G)$.

This in turn implies the following: If $f_1,\dots,f_r$ is a homogeneous system of parameters for $K[V]^G$, then $K[V]^G$ is generated as a ring by elements of degree at most $\operatorname{max}(d_1+\cdots+d_r-r,d_1,d_2,\dots,d_r)$ where $d_i=\operatorname{deg}(f_i)$ and $r$ is the Krull dimension.

Is this still true if $V$ is replaced by a $G$-variety $X$? Or at least a sufficiently nice $G$-variety? It seems likely as we are simply moving to a quotient ring of $K[V]^G$ but I would like to know for sure.

Edit: To add more information about the situation I am working with: I am trying to find a degree bound for a specific action $G$ acting on $V$. I'm looking at a quotient ring $K[V]^G/I$ which I know can be realized as $K[X]^G$ for a certain subvariety $X$ of $V$. I am viewing both of these rings with the standard grading.

I know quite a bit about both $K[X]^G$ and $I$ and can thus say some things about $K[V]^G$. Actually, since posting this question, I have found a set of generators for $K[X]^G$. However, in general, I am still interested in knowing if a degree bound for $K[X]^G$ can be found if the a set of equations defining the null cone are known/a homogeneous system of parameters of $K[X]^G$ is known.

But I must clarify that I am assuming that $K[X]^G$ is a specific quotient ring of $K[V]^G$ with the standard gradings.