Polynomial-time solvability of integer knapsacks in the diagonal Frobenius cone

Let $A$ be an integer matrix with columns ${\boldsymbol v}_1,\ldots,{\boldsymbol v}_n\in\mathbb{Z}^m$, let

$$C=\{\mu_1{\boldsymbol v}_1+\cdots+\mu_n{\boldsymbol v}_n:\mu_1,\ldots,\mu_n\geq 0\},$$

and set ${\boldsymbol v}={\boldsymbol v}_1+\cdots+{\boldsymbol v}_n$. Let $P(A,{\boldsymbol b})$ denote the integer knapsack polytope, and let $\mathrm{g}(A)$ be the diagonal Frobenius number, namely the minimal $s\geq 0$ such that every ${\boldsymbol b}\in(s{\boldsymbol v}+C)\cap\mathbb{Z}^m$ makes $P(A,{\boldsymbol b})$ contain an integer point. Polynomial-time solvability conjecture. The problem is solvable in polynomial time for all instances $(A,{\boldsymbol b})$ with

$${\boldsymbol b}\in(\mathrm{g}(A){\boldsymbol v}+C)\cap\mathbb{Z}^m.$$

The feasible translated cone guarantees that the knapsack polytope contains an integer point for every right-hand side in the specified region, and the conjecture asks whether this structural promise yields a polynomial-time algorithm. The source provides no resolution, so the conjecture remains open.