$⊕P$-completeness of $⊕2SAT$ Is $⊕2SAT$ - the parity of the number of solutions of $2$-$CNF$ formulae $\oplus P$ complete?

This is listed as an open problem in Valiant's 2005 paper https://link.springer.com/content/pdf/10.1007%2F11533719.pdf. Has this been resolved?

Is there any consequence if $⊕2SAT\in P$?
It is shown to be $\oplus P$-complete by Faben:
https://arxiv.org/abs/0809.1836
See Thm 3.5. Note that counting independent sets is same as counting solutions to monotone 2CNF.