The bounded-entry tree-structured integer-programming conjecture

Fix a positive integer $d$. Let $A$ be a matrix with exactly two nonzero entries per row, let $G(A)$ be its associated graph, and suppose that $G(A)$ is a tree and $\|A\|_\infty\leq d$. For integer vectors $w,b$ and bounds $\ell,u$, consider the integer program

$$\max\{w^Tx: Ax \leq b,\ x \in [\ell,u]\cap\mathbb{Z}^n\}.$$

Bounded-entry tree integer-programming conjecture. The integer program can be solved in polynomial time for constant $d$.

This is presented as a simpler unresolved case of polynomial-time solvability for bounded-entry matrices with tree structure. The supplied text does not give a resolution status beyond stating that it is not known.