The generator conjecture for regular local rings

Let $A$ be a commutative noetherian ring with unit and let $0\leq p\leq\dim A$. Denote by $\operatorname{\mathcal{M}}^p_A$ the category of finitely generated $A$-modules whose support has codimension at least $p$. A sequence $f_1,\ldots,f_q$ in $A$ is an $A$-regular sequence if every $f_i$ is a nonunit, $f_1$ is not a zero divisor of $A$, and $f_{i+1}$ is not a zero divisor of $A/(f_1,\ldots,f_i)$ for $1\leq i\leq q-1$. Generator conjecture. For any commutative regular local ring $A$ and any natural number $0\leq p\leq\dim A$, the Grothendieck group $K_0(\operatorname{\mathcal{M}}^p_A)$ is generated by the cyclic modules $A/(f_1,\ldots,f_p)$ where $f_1,\ldots,f_p$ is an $A$-regular sequence. This is a central problem in commutative algebra and algebraic $K$-theory, with connections to Serre's intersection multiplicity conjecture; the source provides no resolution, so the conjecture is recorded as open.