The polynomial-time solvability conjecture for totally -modular integer programs

Let $A\in\mathbb{Z}^{m\times n}$ be a totally $\Delta$-modular matrix, meaning that every square submatrix of $A$ has determinant bounded by $\Delta$ in absolute value. For integer vectors $b,w$ and integer variable vector $x$, consider the integer program

$$\max\{w^Tx: Ax \leq b,\ x \in \mathbb{Z}^n\}.$$

Totally $\Delta$-modular integer-program conjecture. For any constant $\Delta$, this integer program can be solved in polynomial time when $A$ is totally $\Delta$-modular.

The case $\Delta=1$ is the totally unimodular case. Artmann, Weismantel, and Zenklusen answered the conjecture affirmatively for $\Delta=2$, while the cases $\Delta\geq 3$ remain open.