Treglown's conjecture for squares of Hamiltonian cycles in oriented graphs

Let $G$ be an oriented graph, meaning a directed graph with no 2-cycles, on $n$ vertices. Let $\delta^0(G)$ denote its minimum semi-degree, the minimum of its minimum in-degree and minimum out-degree. Treglown's conjecture. If

$$\delta^0(G)\geq\frac{5n}{12},$$

then $G$ contains the square of a hamiltonian cycle. This is an oriented-graph analogue of Pósa's conjecture, following a sufficiently-large-order oriented version of Dirac's theorem; the assertion itself is presented as an open conjecture.