The density conjecture for generators of projective algebras

Let $R$ be the ring of algebraic integers in a number field $K$, and let $A$ be an $R$-algebra that is finitely generated and projective as an $R$-module. For each maximal ideal $\frak{p}$ of $R$, write $\text{\rm N}(\frak{p})$ for the cardinality of $R/\frak{p}$ and, for a positive integer $k$, write $\text{\rm g}_k(A,\frak{p})=\text{\rm g}_k(A/\frak{p}A,R/\frak{p})$. Consider the product

$$\prod_{\frak{p}\in \text{\rm ~m-Spec~} R}\frac{\text{\rm g}_k(A,\frak{p})}{\text{\rm N}(\frak{p})^{mk}},$$

where $m$ is the rank in the free case described by Theorem~. The density conjecture. If this product is positive for some positive integer $k$, then $A$ can be generated by $k$ elements as an $R$-algebra. The product is the natural extension of the density formula known when $A$ is free; the conjecture asserts that its positivity still detects algebra generation when $A$ is merely projective, although no suitable density has been defined in that case.