Preimage preservation of NBOn under onto monotone maps

Establish whether the following holds: for an onto monotone mapping f: X -> Y between continua and any n in the natural numbers, if A belongs to NBOn(Y) (i.e., Y - A is n-Qo), then f^{-1}(A) belongs to NBOn(X) (i.e., X - f^{-1}(A) is n-Qo).

Background

The paper analyzes how various mappings (onto, open, monotone) preserve different hyperspaces of non-n-cut sets under images and preimages. Proposition 4.5 shows that preimages under onto monotone maps preserve CCn, NWCn, and NCn. For NBOn, the authors show preservation under onto and open maps (Proposition 4.4) but do not settle the monotone preimage case.

Resolving this would clarify the behavior of NBOn under common mapping classes, complementing known preservation results for other hyperspaces.