Surjectivity conjecture for primitive elements of additive-type group schemes

Let $A$ be the base ring, let $q$ be a power of the characteristic prime $p$, and let $G\in\mathbb{F}_{q}\text{-}\operatorname{Gr}^{+}_{A}$. Suppose that there is a closed embedding

$$G\hookrightarrow\mathbb{G}_a^N$$

for some $N\in\mathbb{N}$. The functor $\mathcal{M}_p$ assigns to such a group scheme its module of primitive elements of the relevant $\mathbb{F}_p$-weight, and $A[F]$ denotes the corresponding skew polynomial ring. Surjectivity conjecture. The induced morphism

$$\mathcal{M}_p(\mathbb{G}_a^N)=A[F]^N\longrightarrow\mathcal{M}_p(G)$$

is surjective.

This is posed as an open question needed for the analogue of the main equivalence theorem for group schemes locally of finite presentation. It concerns finite generation of the primitive-element module under an additive embedding.