# When is hypertree width more useful than generalized hypertree width?

In analysis of CSPs, there are three width notions that are analogous to treewidth: hypertree width (hw), generalized hypertree width (ghw) and fractional hypertree width (fhw). Moreover the inequalities $$\text{fhw} \le \text{ghw} \le \text{hw}$$ are known. The only motivation for hw instead of ghw that I have seen is that there is an XP algorithm for computing hw, but not for computing ghw. Is there any other motivation for hw instead of ghw? More concretely:

What algorithmic results there are that work when a hypertree decomposition of the smallest width is given, but not when a generalized hypertreewidth decomposition of the smallest width is given?

Every XP/FPT algorithm parametrized by htw also gives an XP/FPT algorithm parametrized by ghtw (provided that the decomposition is given in the input) since there is a linear relation between ghw and hw : $$\mathsf{hw} \leq 3\mathsf{ghw}+1$$. So basically, you can see hw as a easier to compute constant approximation of ghw.
Indeed, if the algorithm is XP for hw, then it is polynomial on $$\alpha$$-acyclic hypergraphs, that is, hypergraphs of hw $$1$$. Now, if you perform the join of the guards inside each bag of the (fractional / generalized) hw decomposition then you get a new CSP / Databases that is $$\alpha$$-acyclic. Moreover, each new relation is of XP size in the original DB/CSP. Now if you can rewrite your problem on this new database, this directly give you an XP algorithm for fhw.