Faben and Jerrum's homomorphism-counting dichotomy conjecture for graphs

Let $H$ be a (not necessarily simple) graph. The involution-free reduction of $H$ is the lexicographically minimal involution-free graph $H'$ such that $H\rightarrow^* H'$. The problem $\oplus \operatorname{HomsTo}(H)$ asks for the number of graph homomorphisms to $H$, modulo $2$.

Faben and Jerrum's conjecture. The problem $\oplus \operatorname{HomsTo}(H)$ is in $\mathrm{FP}$ if the involution-free reduction of $H$ is the empty graph, a singleton vertex with or without a self-loop, or a graph with two isolated vertices, exactly one of which has a self-loop. Otherwise, it is $\oplus\mathrm{P}$-complete.

This conjecture proposes a complexity dichotomy for counting graph homomorphisms modulo $2$ on arbitrary graphs, extending the known dichotomy for trees. The supplied text does not state whether the conjecture has been resolved.