Finiteness conjecture for tame integral positive $\operatorname{SL}_3$-friezes of height 4

Let a tame integral positive $\operatorname{SL}_3$-frieze be a tame frieze pattern whose entries are nonnegative integers and whose adjacent $3\times3$ minors have determinant $1$; its height is the number of nontrivial rows. Finiteness conjecture for tame integral positive $\operatorname{SL}_3$-friezes. There are only finitely many tame integral positive $\operatorname{SL}_3$-friezes of height $4$. More precisely, the number of such friezes could be $26952$. The preceding discussion explains that tame friezes are periodic and are determined by a periodic sequence of tuples, so the conjecture asks for finiteness—and suggests the possible count $26952$—in this specific height-four case.