Planar Exact Cover by even-size sets
The problem seems to be NP-complete. My reduction is from Planar 1-in-3-SAT, just like in their paper. For each variable, take $2k$ vertices. Each of these is covered by two 4-sets and each 4-set covers two consecutive vertices. Thus, we need to take every second of the 4-sets for each variable, this corresponds to TRUE/FALSE. The other two vertices of the 4-sets are from the clauses. (If the number of negated and unnegated occurrences of some variable differs, then convert some of the 4-sets into 2-sets.) Each clause has 4 vertices; one in the center and the other three in a triangle. Any two vertices of the triangle are contained in a 4-set that belongs to its appropriate literal. The central vertex is in three 2-sets, one for each vertex of the triangle. Not sure how clear this was, so I've made a sketch:

https://photos.app.goo.gl/TMc49pfeT6RZA9Vb8

It is easy to see that this hypergraph is indeed planar.
The proof of the reduction is also quite straight-forward.
Let me know if you have any questions.