Generalized hypertreewidth is a generalization of treewidth to hypergraphs. Unlike treewidth, it is not tractable, for a fixed width $k \in \mathbb{N}$, given a hypergraph $H$, to determine if $H$ has treewidth $\leq k$. Indeed, it was recently shown (see also TCS.SE question) that intractability holds already for $k=2$: it is NP-hard, given an input hypergraph $H$, to determine if $H$ has generalized hypertreewidth $\leq 2$.
However, this hardness proof uses hypergraphs of unbounded arities (the arity of a hyperedge is simply its cardinality). My question: is the same true when the arity of the hyperedges of the input graph is bounded by a constant? In fact, I do not know what happens already in the case of graphs: for a fixed $k\in\mathbb{N}$, can we determine in PTIME, given a graph $G$, whether $G$ has generalized hypertreewidth $\leq k$?
Of course, when the arity is bounded to a constant $a \in \mathbb{N}$, then generalized hypertreewidth is obviously related to the treewidth of the hypergraph (defined as that of its Gaifman graph, i.e., each hyperedge is encoded as a clique). Specifically, the generalized hypertreewidth is less than the treewidth and is at least the treewidth divided by $a$. So one approach for tractability (suggested to me by Stefan Mengel) would be to first check in PTIME if the treewidth is $\leq k \times a$: if not then the generalized hypertreewidth cannot be $\leq k$, and if yes, maybe one can compute the generalized hypertreewidth exactly in PTIME using the fact that the graph has bounded treewidth. So one related question is: for any fixed $a \in \mathbb{N}$, given a hypergraph $H$ with edges having arity at most $a$, is it PTIME to compute exactly the generalized hypertreewidth of $H$?
