The infinitude conjecture for observable non-reductive subgroups

Let $G$ be an algebraic group and $H\subseteq G$ an observable subgroup. Write $W(H)=N_G(H)/H$ for the corresponding quotient of the normalizer. The subgroup $H$ is reductive if it has trivial unipotent radical. Infinitude conjecture. If $H$ is observable but not reductive, then $W(H)$ is infinite. This conjecture would characterize a structural feature of quasiaffine but non-affine homogeneous spaces beyond the reductive case; the source gives no resolution.