Finiteness of inscribed rectangles in plane separating continua

Determine whether there exists a plane separating continuum for which every embedding admits only a countable, or even a finite, number of inscribed rectangles.

Background

The paper proves existence of at least one inscribed rectangle for every embedding of every plane separating continuum, but does not analyze the cardinality of the collection of such rectangles. The question asks whether that collection can be countable or finite for some plane separating continuum.