Overring noetherianity conjecture for graded maximal orders

Let $T$ be the algebra in the paper's main theorems, let $Q_{\operatorname{gr}}(T)$ denote its graded quotient ring, and consider graded overrings $T\subseteq R\subset Q_{\operatorname{gr}}(T)$ and the corresponding algebra $A$. Overring noetherianity conjecture. Theorem~$(1)$ holds even when $R$, respectively $A$, is not assumed to be noetherian. In particular, finitely graded maximal orders $T\subseteq R\subset Q_{\operatorname{gr}}(T)$ are automatically noetherian; analogous assertions hold for overrings of $S$. The paper proves the corresponding results under noetherian hypotheses, while this extension remains open.