Shiva Kintali has just announced a (cool!) result that graph isomorphism for bounded treewidth graphs of width $\geq 4$ is $\oplus L$-hard. Informally, my question is, "How hard is that?"
We know that nonuniformly $NL \subseteq \oplus L$, see the answers to this question. We also know that it is unlikely that $\oplus L = P$, see the answers to this question. How surprising would it be if $L=\oplus L$? I have heard many people say that $L=NL$ would not be shocking the way $P=NP$ would.
What are the consequences of $L=\oplus L$?
Definition: $\oplus L$ is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero number of acceptance paths), and which is further restricted to work in logarithmic space.