NP-hardness of a planar SAT variant
The following paper answers the question in the affirmative – the variant remains NP-hard using a reduction from Monotone Planar 3-SAT:
http://epubs.siam.org/doi/abs/10.1137/1.9781611976465.105
(arXiv: http://arxiv.org/abs/2009.12369)
The paper presents a slightly more restricted variant, Monotone Planar 3-SAT with Neighboring Variable Pairs, which requires each clause with three variables to contain two variables that are adjacent on the line on which all variables are embedded.
If there is a bi-enclosing clause in this variant, it must enclose a redundant clause due to monotonicity, so we can assume no bi-enclosing clauses.