Conjecture on piecewise linear Seshadri functions for non-simple abelian surfaces

Let $A$ be a non-simple abelian surface with Picard number two. The Seshadri function of $A$ is piecewise linear precisely in the cases represented by the intersection matrices in the stated example:

$$\left(\begin{array}{cc}2&2\\2&0\end{array}\right),\qquad \left(\begin{array}{cc}2&3\\3&0\end{array}\right),\qquad \left(\begin{array}{cc}0&n\ &0\end{array}\right)\quad\text{for }n\in\{1,\dots,6\}.$$

Piecewise linearity conjecture. The Seshadri function of $A$ is piecewise linear precisely in these cases.

The listed intersection matrices provide known examples of piecewise linear Seshadri functions. The source states that it is unknown whether other cases exist, while computer-assisted computations suggest that no others do; the precise classification remains open.