The disjoint-path comparison conjecture for graded graphs

Let $G$ be a graded graph with root $\rho$, levels $V_0,\ldots,V_N$, and IID vertex labels with common distribution $\mu$. For an event $B$ of label sequences, let $P(G;B)$ be the probability that some graded path in $G$ has its label sequence in $B$. Let $M$ be the number of distinct graded paths in $G$, and let $G'$ consist of $M$ paths of length $N$ that are disjoint except at $\rho$. Disjoint-path comparison conjecture.

$$P(G;B)\leq P(G';B).$$

The comparison is known for graded graphs arising from trees, while the source reports only special cases for general graded graphs, including certain cases with $N=2$.