Integrality conjecture for the linear program on directed graphs

Let $G=(V,D)$ be a directed graph, let $G_1$ be a subgraph of $G$, and let $Z$, $P$, and $Y_a$ be the vertex sets used to define the linear program $\operatorname{Lp}(G,G_1,Z,P,Y_a)$. The program has optimal value $|Z|+|P|$.

Integrality conjecture. If the optimal value of $\operatorname{Lp}(G,G_1,Z,P,Y_a)$ is $|Z|+|P|$, then there is an integer solution.

The authors report that this implication held in all their computations. If true in general, it would allow the polynomial-time linear program to replace the corresponding integer linear program, which is NP-complete.