# Best known bounds on feedback arcset in high-girth directed graphs?

I asked this question over at MathOverflow, but thinking about it a little more I think it is a more natural fit here.

Let $$G$$ be a directed graph with $$n$$ vertices and $$m$$ edges such that every directed cycle in $$G$$ has length at least $$m/k$$. An arcset of $$G$$ is defined as a set of edges $$X$$ whose removal from $$G$$ leaves an acyclic graph. What is the best known bound on the size of the smallest such $$X$$? The best I have been able to find in the literature is a bound $$|X| \leq O(k \log k \log \log k)$$ due to Seymour (see 1.4 here: https://link.springer.com/article/10.1007/BF01200760 ).

Meanwhile, the best known worst-case lower bound is $$|X| \geq O(k \log k)$$, which can be obtained by taking an undirected expander graph and orienting the edges appropriately. Has this gap of $$\log \log k$$ been closed?