The generalized center decomposition conjecture for derived planar nearrings

Let $F$ be a nearfield with kern $K$, and let $V$ be a finite-dimensional $F$-nearvector space defining a derived planar nearring. Suppose that

$$V=V_1\oplus\dots\oplus V_n$$

is its regular decomposition, with $V_i=F^{n_i}$.

Generalized center decomposition conjecture. The generalized center satisfies

$$D(V)=D(V_1)\oplus\dots\oplus D(V_n),$$

with

$$D(V_i)=K^{n_i}.$$

This predicts that the generalized center of the derived planar nearring decomposes according to the regular components of the underlying nearvector space, with each component determined by the kern of the nearfield. The supplied text gives examples supporting this behavior, but does not state that the general claim has been proved.