Morell and Skutella's two-sided convex-combination conjecture for acyclic single-source flows

Consider the single-source unsplittable flow problem on an acyclic directed graph, with common source $s$, commodities ending at sinks $t_i$ with demands $d_i$, and a fractional flow $(x_e)_{e\in E}$. An unsplittable flow $(y_e)_{e\in E}$ is induced by one $s$--$t_i$ path for each commodity, and let $d_{\max}\coloneqq\max_{i\in[k]}d_i$. Define the upper and lower bounds

$$y_e\leq x_e+d_{\max}\qquad\text{for every arc }e\in E,$$

and

$$y_e\geq x_e-d_{\max}\qquad\text{for every arc }e\in E.$$

Morell and Skutella's conjecture. Any fractional flow $(x_e)_{e\in E}$ can be expressed as a convex combination of unsplittable flows $(y_e)_{e\in E}$ satisfying both bounds. The supplied text states that the analogous existence result for a single unsplittable flow is known for acyclic planar digraphs, but does not state a resolution of this convex-combination strengthening.