Let $X$ be a scheme (If necessary assume $X$ is separated and finite type) over a ring $R$. Let $L$ be a very ample invertible sheaf on $X$, as defined in definition 29.38.1 in the Stacks project. Is the graded ring $\oplus_{n \geq 0} \Gamma(X, L^{\otimes n})$ finitely generated in degree 1 over degree 0?
I tried some examples : if $X = \mathbb{P}^n_R$ and $L = O(1)$, then $\Gamma(X, L^{\otimes n})$ is just a polynomial ring. By theorem 29.39.1, there is an immersion $i : X \rightarrow \mathbb{P}^n_R$ with $L = i^* O(1)$. I tried to use that to reduce the general case to the $X = \mathbb{P}^n_R$ case. But since $i$ is just an immersion, I can't figure out how to relate $\Gamma(\mathbb{P}^n_R, O(n))$ with $\Gamma(X, L^{\otimes n})$ .
I did some searching and found this question but the counterexample involves a non-torsion invertible sheaf on an elliptic curve, so it can't be very ample.