In the paper "Hard Tiling Problems with Simple Tiles", Moore and Robson prove that Cubic Planar Positive 1-in-3 Sat in NP-complete by a reduction from Positive 1-in-3 Sat.
Cubic Planar Positive 1-in-3 Satisfiability
Instance: A boolean formula $B=(X,C)$ where $X$ is a set of variables, $C$ a set of 3-element subsets over $X$ (clauses), and graph of B is a cubic planar graph.
Question: Does there exist a truth assignment for $X$ such that each clause is $C$ has exactly one true variable?
(see the note of the https://cstheory.stackexchange.com/a/42605/47855 regarding terminology)
Then they prove that a particular tiling problem is NP-complete by a reduction from Cubic Planar Positive 1-in-3 Sat. The following note is given after their reduction:
While the reduction from this version of SAT to the tiling problem we gave in the previous section is parsimonious, this does not show that the counting problem is #P-complete, since we do not have such a result for Cubic Planar Positive 1-in-3 SAT. We leave this as an open problem for the reader.
This gave me the impression that the counting version of 1-in-3 Sat is indeed #P-complete, but I couldn't find any source claiming this. Is the counting version of 1-in-3 Sat #P-complete?