Cycle-factor extension of the oriented discrepancy conjecture

Let $G$ be an oriented graph, that is, a loopless directed graph with at most one edge between any two vertices. Let $\delta(G)$ denote its minimum total degree. Let $n=n_1+n_2+\cdots+n_k$ with $n_i\geq 3$ for each $1\leq i\leq k$. A cycle factor $\mathcal{C}$ is a collection of vertex-disjoint cycles covering all vertices of $G$; let $\sigma_{\max}(\mathcal{C})$ denote the maximum number of edges of $\mathcal{C}$ in either consistent traversal direction.

Cycle-factor extension conjecture. If

$$\delta(G)\geq\sum_{i=1}^k\left\lceil\frac{n_i}{2}\right\rceil,$$

then $G$ contains a cycle factor $\mathcal{C}$ satisfying

$$\sigma_{\max}(\mathcal{C})\geq\delta(G).$$

This conjecture extends the Hamilton-cycle statement to cycle factors and is presented as an extension of the El-Zahar conjecture. The supplied text identifies it as open, while noting that determining the precise value of $\sigma_{\max}(C)$ in the Hamilton-cycle setting also remains open.