Fuchsian Nakayama category classification conjecture

A connected perpendicular subcategory is a connected perpendicular subcategory of a Fuchsian singularity category, and a Nakayama category $\mathcal{N}_n(r)$ is a Nakayama category with the indicated parameters. A category is piecewise hereditary when it is derived equivalent to the derived category of a hereditary abelian category.

Fuchsian-category conjecture. The following properties hold:

• (F1) Each connected perpendicular subcategory of a Fuchsian singularity category is either Fuchsian or piecewise hereditary.
• (F2) If a Nakayama category $\mathcal{N}_n(r)$ is Fuchsian, then $\mathcal{N}_{n-1}(r)$ is either Fuchsian or piecewise hereditary.

These properties are proposed as principles for classifying Fuchsian Nakayama categories. The source states them as conjectural and provides no resolution.