Conjecture on the exceptional set of Seshadri functions in Picard number two

Let $A$ be an abelian surface with Picard number two, and let $Z$ be the subset of the compact cross section $[-1,1]$ on which the restricted Seshadri function is not locally piecewise linear.

Exceptional-set conjecture.

1. If $A$ is simple, then $Z$ is perfect.
2. If $A$ is non-simple, then $Z$ is either perfect or finite with $|Z|\in\{0,1,2\}$. Moreover, there are only finitely many intersection matrices such that $Z$ is finite.

The set $Z$ measures the complexity of the Seshadri function: it is empty in globally piecewise linear cases, while computational evidence suggests the stated perfect-or-finite dichotomy. The assertions are presented as conjectural computational evidence and remain open.