Finiteness conjecture for extendable mapping classes of embedded surfaces

Let $F_g$ be a closed orientable surface of genus $g$ embedded in $S^4$. An orientation-preserving self-homeomorphism of $F_g$ is extendable if it extends to a homeomorphism of the pair $(S^4,F_g)$; its mapping class is the corresponding element of the mapping class group $\operatorname{MCG}(F_g)$. Finiteness conjecture. There is some surface $F_g$ embedded in $S^4$ such that it has finitely many mapping classes of extendable self-homeomorphisms. The preceding norm construction controls the induced homological action, but for genus greater than one it does not control the Torelli group, so finiteness at the level of the full mapping class group remains open.