The period-set containment conjecture for max-type equations

For an integer $k\geq 5$, consider Equation (Gk) and let $\mathrm{Per}(F_k)$ be its set of periods.

Period-set containment conjecture.

$$\mathrm{Per}(F_k) \subset \left\{1, \frac{1}{2}(3-(-1)^k), 2k, 3k-1\right\} \cup \left\{(3k-2)\cdot a+(3k-1)\cdot b \mid \gcd(a,b)=1\right\}.$$

The conjecture is motivated by computer simulations for $k=5$ and $k=6$, together with the periodicity results established earlier in the paper. It predicts that, beyond the explicitly listed periods, every period is a coprime linear combination of $3k-2$ and $3k-1$; the general description of $\mathrm{Per}(F_k)$ remains open.