Growth-constant bound for height-one-growth prime ideals

Let $k$ be a field and let $A$ be a finitely generated non-PI prime Noetherian $k$-algebra of quadratic growth. Write

$${\rm GC}(A)=\inf_V\limsup_{n\rightarrow\infty}\frac{\dim(V^n)}{n^2},$$

where the infimum is over all frames $V$ of $A$.

Growth-constant bound conjecture. There exists a function $F:(0,\infty)\rightarrow(0,\infty)$ such that $A$ has at most $F({\rm GC}(A))$ prime ideals $P$ satisfying ${\rm GKdim}(A/P)=1$.

The preceding theorem proves finiteness, and gives an explicit bound in the monomial case; the conjecture asks for a bound depending only on the growth constant for all finitely generated non-PI prime Noetherian algebras of quadratic growth.