The odd-order-free Fitting-height bound for coprime factorisations

Let $G=AB$ be a finite soluble group factorised by its proper subgroups $A$ and $B$ with $\gcd(|A|,|B|)=1$. Here $h(X)$ denotes the Fitting height of a soluble group $X$, and $d(X)$ its derived length. Fitting-height bound conjecture. One has

$$h(G)\leq h(A)+h(B)+2d(B)-1.$$

The conjecture removes the odd-order hypothesis on $B$ from the preceding theorem; the paper explains that this hypothesis enters through the use of a theorem of Kazarin and is believed to be only technical. The result would extend the established bound to arbitrary coprime factorisations of finite soluble groups.