Constant discrepancy conjecture for random regular hypergraphs

Let $t>0$ be an integer, and let $\mathcal{H}$ be a random $t$-regular hypergraph on $n$ vertices and $K\frac{n}{\log n}$ edges, where $K>0$ is an absolute constant. Write $\operatorname{disc}(\mathcal{H})$ for the hypergraph discrepancy.

Constant discrepancy conjecture. There is an absolute constant $K>0$ such that, for every integer $t>0$, a random $t$-regular hypergraph $\mathcal{H}$ on $n$ vertices and $K\frac{n}{\log n}$ edges satisfies

$$\operatorname{disc}(\mathcal{H})=O(1)$$

with high probability.

The conjecture concerns the regime where the number of edges is on the order of $n/\log n$, where the authors believe a phase transition for constant discrepancy occurs. Existing results establish constant discrepancy in related random-hypergraph models and denser regimes, but the stated random regular case remains open.