Bang-Jensen–Bessy–Thomassé conjecture on cycles and girth

Let $k$ and $g$ be positive integers, and let $f(k,g)$ be the minimum integer such that every finite simple digraph of girth $g$ and minimum outdegree at least $f(k,g)$ contains $k$ vertex-disjoint directed cycles. The circular digraph construction described in the source gives $f(k,g)\geq \left\lceil\frac{g}{g-1}k\right\rceil$. Bang-Jensen–Bessy–Thomassé conjecture.

$$f(k,g)=\left\lceil\frac{g}{g-1}k\right\rceil.$$

The conjecture is presented as a proposed strengthening in terms of girth, but this paper disproves it, so the conjecture is refuted.