2CNF with 3 variable occurences If we do not allow unit clauses, can 2CNF containing 3 occurences per variable be unsatisfiable?

Every time I try to connect opposing variables to a cycle, I need to draw 4 edges for at least one variable (2 per positive and negative occurences).

So, is it even possible?
The following is unsatisfiable:
$$
(x \lor y) \land (x \lor \lnot y) \land (\lnot x \lor z) \land (\lnot z \lor w) \land (\lnot z \lor \lnot w) \land (y \lor w).
$$
This contains every variable exactly three times, not all of them of the same polarity. If you allow variables to appear less than three times, you can drop the last clause $y \lor w$.