CSPs with unbounded fractional hypertree width
You linked to two papers, both with conjectures.  I presume you mean Grohe's 2007 conjecture.

This question was answered in 2008:


  Theorem 5. CSP(C$_0$,_) is in NP, but neither in P nor NP-complete (unless P = NP).  Moreover, the set C$_0$ can be decided in deterministic polynomial time.



Manuel Bodirsky and Martin Grohe, Non-dichotomies in Constraint Satisfaction Complexity, ICALP 2008.  doi:10.1007/978-3-540-70583-3_16 (PDF)


The idea is to blow holes in the instance sizes of CLIQUE, by the same delayed diagonalization technique introduced by Ladner for his theorem.  Note that the set C$_0$ contains arbitrarily large cliques, and the fractional hypertree width of an $n$-clique is $n/2$.  So it is possible to have CSPs of the form CSP(A,_) that are of intermediate complexity, where A has unbounded fractional hypertree width.  This answers Grohe's conjecture in the negative.

At the same conference Chen, Thurley, and Weyer had a paper with a similar result, but that required strong embeddings so technically wasn't of the right form for the conjecture.


Yijia Chen, Marc Thurley, and Mark Weyer, Understanding the Complexity of Induced Subgraph Isomorphisms, ICALP 2008.  doi:10.1007/978-3-540-70575-8_48 (PDF)


Finally, any class of CSP instances can be transformed into a representation with worst-case fractional hypertree width.  In many cases this transformation is polynomially bounded in size and can be done in polynomial time.  This means that it is easy to generate CSPs with unbounded fractional hypertree width, even modulo homomorphic equivalence.  These CSPs are not going to be of the form CSP(A,_) since the target structures are special, but they do provide a neat reason why the CSPs defined by restricting just the source structures are not all that interesting: it is often just too easy to hide the tree-like structure of a CSP instance by changing the representation so that the source structure has large width.  (This is discussed in chapter 7 of my thesis.)