Restriction of purely periodic least periods to pair sums

Establish that every purely periodic non-additive three-move subtraction game with a>=2 and gcd(a,b,c)=1 has least Grundy-value period belonging to the set {a+b,c+a,c+b}.

Background

The paper proves pure periodicity and exact least periods for the admissible regions identified by its rotation criterion. For the inert period a+b, necessity is established when c>=2(a+b), but the authors do not prove that all purely periodic games with larger periods must use c+a or c+b.

This conjecture isolates the period-size component of the broader necessity classification. The authors explicitly identify as unresolved the possibility of a purely periodic non-additive game with least period exceeding a+b but with a period outside the two remaining candidate sums, as well as cases with b<c<2(a+b).