Nontrivial strict inclusions among right kernels

Construct an example, for an open surface over a local field, in which the right kernel of the Brauer–Manin pairing is strictly intermediate between the prime-to-residue-characteristic Brauer groups associated with a compactification and a regular model, with all three groups nonzero and pairwise distinct.

Background

The paper gives examples showing that the two inclusions bounding the right kernel can be strict in general, and another example in which the right kernel agrees with one of the boundary groups. These examples do not realize the configuration in which both inclusions are strict while the three groups are all nonzero.

The authors explicitly pose the construction of such an example as a question and subsequently state that a positive answer is believable, proposing a strategy under additional conditions.