Complexity conjecture for unrestricted shadow inflection minimization

An embedded shadow is a combinatorial model of a plane curve shadow, and \textnormal{\textsc{MinInflection}} denotes the decision problem asking whether the minimum inflection number is at most a prescribed bound. A building polygon has bounded degree when its degree is bounded uniformly across the shadow, while the cycle rank measures the number of independent cycles in the underlying combinatorial structure.

Complexity conjecture. For an appropriate purely combinatorial encoding of embedded shadows, the decision problem $\textnormal{\textsc{MinInflection}}$ for unrestricted shadows is NP-hard. The same should already hold for shadows for which all building polygons have uniformly bounded degree, provided the cycle rank is allowed to grow.

The preceding bounded-degree results give polynomial-time or fixed-parameter algorithms for restricted tree-like and tree--necklace classes, so this conjecture predicts that NP-hardness appears only after substantially enlarging the class or allowing its cycle rank to grow.