Jaeger orientations in random regular graphs

Let $\mathcal{G}_{n,d}$ be the uniform model of random $d$-regular graphs on the vertex set $[n]=\{1,2,\ldots,n\}$. A $p$-orientation is an orientation of a $d$-regular graph in which every vertex has in-degree $p$ or out-degree $p$. An event holds asymptotically almost surely (a.a.s.) if its probability tends to $1$ as $n\to\infty$. Jaeger-orientation conjecture. For every fixed $p\geq 1$, a.a.s. the random graph $\mathcal{G}_{n,4p+1}$ has a Jaeger orientation.

This extends the known a.a.s. results for $p=1$, $p=2$, and all sufficiently large $p$. The conjecture concerns the typical existence of these orientations even though the deterministic Jaeger conjecture is false for $p\geq3$.