Positivity conjecture for frieze patterns of type $\Lambda_{p_1,\ldots,p_s}$

Let a frieze pattern be of Type $\Lambda_{p_1,\ldots,p_s}$, and let the realizability test be the criterion given in Proposition. A frieze pattern of Type $\Lambda_{p_1,\ldots,p_s}$ is understood in the sense of the paper, and a quotient dissection is a dissection whose associated frieze pattern is obtained by the quotient construction.

Positivity conjecture. The only frieze patterns of Type $\Lambda_{p_1,\ldots,p_s}$ with negative entries are those which fail the realizability test. In particular, all quotient dissections yield frieze patterns with positive entries.

This conjecture proposes that failure of the realizability test exactly accounts for negative entries, extending the positivity result proved under the sufficient condition in Proposition. The examples preceding the conjecture show unrealizable patterns with both positive and negative entries, while the general assertion for all quotient dissections remains open in the supplied text.