The Not All Equal $k$-SAT problem (NAE $k$-SAT), given a set $C$ of clauses over a set $X$ of boolean variables such that each clause contains at most $k$ literals, asks whether there exists a truth assignment of the variables such that each clause contains at least one true and at least one false literal.
The PLANAR NAE $k$-SAT problem is the restriction of NAE $k$-SAT to those instances where the incidence bipartite graph of $C$ and $X$ (i.e. the graph of parts $C$ and $X$ with an edge between $x\in X$ and $c\in C$ if and only if $x$ or $\overline{x}$ belongs to $c$), is planar.
It is known that NAE 3-SAT is NP-complete (Garey and Johnson, Computers and Intractability; A Guide to the Theory of NP-Completeness), but PLANAR NAE 3-SAT is in P (see Planar NAE3SAT is in P, B. Moret, ACM SIGACT News, Volume 19 Issue 2, Summer 1988 - unfortunately I do not have access to this paper).
Is PLANAR NAE $k$-SAT in P for some $k\geq 4$? Is there a value of $k$ for which it has been shown to be NP-complete?