Powell's Conjecture on the Goeritz group of $S^3$ is stably true

In 1980 J. Powell proposed that, for every genus $g$, five specific elements suffice to generate the Goeritz group ${\mathcal{G}}_g$ of genus $g$ Heegaard splittings of $S^3$. Powell’s Conjecture remains undecided for $g\ge 4$. Let ${\mathcal{P}}_g\subset {\mathcal{G}}_g$ denote the subgroup generated by Powell’s elements. Here we show that, for each genus $g$, the natural function ${\mathcal{G}}_g\to {\mathcal{G}}_{g+1}\mathrm{/}{\mathcal{P}}_{g+1}$ is trivial.