The component-degree conjecture for the associated map of quadratic APN functions

Let $F$ be a quadratic APN function of $n$ variables, with $n\geq3$, and let $\Phi_F:\mathbb{F}_2^n\to\mathbb{F}_2^n$ be its associated map. For $v\in\mathbb{F}_2^n$, a component of $\Phi_F$ is the Boolean function $v\cdot\Phi_F$.

Component-degree conjecture. For every nonzero $v\in\mathbb{F}_2^n$,

$$\deg(v\cdot\Phi_F)=n-2.$$

The conjecture is based on computational experiments. It strengthens the paper's general degree bounds by asserting exact degree for every nonzero component.