Admissibility criterion for regular spherical polygons

Let $P$ be a regular spherical polygon, and let its polar be the corresponding polar spherical polygon. Call $P$ admissible when it has the finite sentinel-set property: there is a finite subset $S$ of $P$ such that, for every spherical isometry $T$, the polygons $P$ and $TP$ intersect if and only if $TP$ contains a point of $S$ or $P$ contains a point of $TS$.

Admissibility criterion. A regular spherical polygon is admissible if and only if the inner angle of its polar is at least

$$\pi/2.$$

The criterion would remove the extra side-length condition currently used in the proof for spherical triangles and is motivated by the necessity of the angle bound and by an extensive search for counterexamples. The statement remains open in the source.