The existence conjecture for recursive APN functions

Let $n\in\mathbb{N}$ with $n\geq 2$. An APN function $F_n\colon\mathbb{F}_{2}^n\to\mathbb{F}_{2}^n$ is recursive if, for each $i\in\{2,\dots,n-1\}$, there exists an APN function $F_i\colon\mathbb{F}_{2}^i\to\mathbb{F}_{2}^i$ such that

$$F_2\prec F_3\prec\dots\prec F_{n-1}\prec F_n.$$

Recursive APN function conjecture. There exists a recursive APN function in every dimension $n\in\mathbb{N}$ with $n\geq 2$.

The conjecture is motivated by examples of quadratic APN functions in dimension $8$ containing APN restrictions successively down to dimension $2$, together with the authors' construction of many new APN extensions. Its resolution is not given in the source.