Kernel-growth conjecture for degree-preserving polynomial derivations

Let $k$ be a field of characteristic zero, and let

$$\Theta \colon k[x_1,\dots,x_N]\to k[x_1,\dots,x_N]$$

be a derivation that sends homogeneous polynomials to homogeneous polynomials of the same total degree. For each $m$, let $K_m$ be the kernel of the restriction of $\Theta$ to the homogeneous polynomials of degree $m$.

Kernel-growth conjecture. The dimension $\dim K_m$ grows polynomially in $m$ of degree at most $N-2$.

This proposes that the polynomial-growth estimate proved for the specific derivations arising from the spectral ball holds for every degree-preserving derivation over a field of characteristic zero. The supplied text gives no resolution, so the conjecture is recorded as open.