Conjecture on the minimum acyclic number of oriented triangle-free graphs

For each positive integer $n$, let $\vec a(n)$ be the minimum of $\vec\alpha(D)$ over all oriented triangle-free graphs $D$ of order $n$, where $\vec\alpha(D)$ denotes the maximum size of an acyclic vertex set. Minimum acyclic number conjecture.

$$\vec a(n)=\Theta\!\left(\sqrt{n\log n}\right).$$

The conjecture is motivated by the triangle-free process and would match the known lower bound up to constant factors, while improving the currently known upper bound asymptotically.