The average Second Neighborhood Conjecture for Eulerian digraphs

Let $D$ be an Eulerian digraph with no loops or digons. For each vertex $v\in V(D)$, let $N^{+}(v)$ and $N^{+2}(v)$ denote its first and second out-neighborhoods, respectively. Average Second Neighborhood Conjecture.

$$\sum_{v\in V(D)}|N^{+2}(v)|\geq \sum_{v\in V(D)}|N^{+}(v)|.$$

Equivalently, the average size of the second out-neighborhood is at least the average size of the first out-neighborhood. The supplied source attributes this conjecture to Conjecture 6.15 of Sullivan and gives no evidence that it has been resolved.