Henk–Weismantel conjecture on Hilbert bases of linear Diophantine equations

For positive integer vectors ${\bf a}=(a_1,\ldots,a_n)\in\mathbb Z_{\geq1}^n$ and ${\bf b}=(b_1,\ldots,b_m)\in\mathbb Z_{\geq1}^m$, let $\mathcal S({\bf a},{\bf b})$ be the set of nonnegative integer solutions $({\bf x},{\bf y})$ to

$$a_1x_1+\cdots+a_nx_n=b_1y_1+\cdots+b_my_m.$$

A nonzero solution is minimal if it cannot be written as the sum of two nonzero solutions, and let $\mathcal H({\bf a},{\bf b})$ be the set of all minimal solutions. For $1\leq i\leq n$ and $1\leq j\leq m$, write ${\bf g}_{i,j}=(b_j{\bf e}_i,a_i{\bf e}_{n+j})$ and define

$$\mathcal W({\bf a},{\bf b})=\operatorname{conv}\bigl(\{{\bf 0}\}\cup\{{\bf g}_{i,j}:1\leq i\leq n,\leq j\leq m\}\bigr).$$

Henk–Weismantel conjecture. Every minimal solution lies in this convex hull:

$$\mathcal H({\bf a},{\bf b})\subseteq\mathcal W({\bf a},{\bf b}).$$

This conjecture concerns the geometry of the Hilbert basis of the pointed rational cone defined by the linear Diophantine equation. It was attributed in the source to Henk and Weismantel, and independently to Hosten and Sturmfels; the supplied text gives no evidence of a resolution.