Degree periodicity conjecture for polynomial maps

Let $L/K$ be a finite extension of number fields and let $m\in\mathbb{N}$. Let

$$f=(f_1(x_1,\ldots,x_m),\ldots,f_m(x_1,\ldots,x_m))\in L[x_1,\ldots,x_m]^m$$

be a polynomial map from $L^m$ to $L^m$. For an element $a=(a_1,\ldots,a_m)\in L^m$, define $\deg_K f^{(n)}(a)$ to be the degree over $K$ of the field obtained by adjoining the coordinates of $f^{(n)}(a)\in L^m$. Degree periodicity conjecture. The sequence

$$\{\deg_K f^{(n)}(a)\}_{n\geq 0}$$

is virtually periodic. This generalizes the preceding one-dimensional theorem; for $m>1$, it is a consequence of the affine $\mathbb{A}^{2m}$ case of the dynamic Mordell–Lang conjecture, which is still open.