Finiteness conjecture for automorphism groups in case c) of Theorem on kernels

Let $D$ be a derivation as in case $c)$ of Theorem, and let ${\rm Aut}(D)$ denote its automorphism group. Finiteness conjecture. In case $c)$ of Theorem, the group ${\rm Aut}(D)$ is finite. The preceding examples establish finiteness for related classes of derivations, but the statement itself is presented as a conjecture for the derivations covered by case $c)$.