Bisection threshold conjecture for $p$-orientations

For each $d\geq3$, let $p^*(d)$ be the minimum value of $p\in\{1,2,\ldots,\lfloor d/2\rfloor\}$ such that a.a.s. the random graph $\mathcal{G}_{n,d}$ has a $p$-orientation. Define

$$p_{\mathrm{bis}}(d):=\min\left\{p\mid 1\leq p<d/2\text{ and a.a.s. }\mathcal{G}_{n,d}\text{ has a bisection of size }\geq(d-2p)n/2\right\}.$$

Here a bisection is a partition of the vertex set into two parts whose sizes differ by at most one, and its size is the number of edges crossing between the parts. Bisection-threshold conjecture. For all $d\geq3$,

$$p^*(d)\in\{p_{\mathrm{bis}}(d),p_{\mathrm{bis}}(d)+1\}.$$

The conjecture proposes that the threshold for a.a.s. existence of $p$-orientations is determined, up to one unit, by the maximum-bisection threshold. It is consistent with the data reported in the paper, while the general assertion remains open.