Fractional arboricity bounded-diameter conjecture

Let $G$ be a graph, let $k\geq1$ be a natural number, and let $\varepsilon>0$ be real. Write $\Upsilon_f(G)$ for the fractional arboricity and $\Upsilon_d(G)$ for the minimum number of forests whose components all have diameter at most $d$.

Fractional arboricity bounded-diameter conjecture. There exists $d(k,\varepsilon)$ such that, if

$$\Upsilon_f(G)\leq k-\varepsilon,$$

then

$$\Upsilon_{d(k,\varepsilon)}(G)\leq k.$$

This is proposed as a strong bounded-diameter consequence of fractional arboricity being separated from the integer arboricity threshold. The paper presents it as an additional open conjecture.