Extension conjecture for step-isometries on dense subsets of Banach spaces

Let $X$ be a strictly convex Banach space with separable dual, let $S$ be a countable dense subset of $X$, and let $T$ be a step-isometry on $S$.

Step-isometry extension conjecture. The map $T$ can be extended to a step-isometry on the whole of $X$.

The paper notes that step-isometries on the whole of any strictly convex space are necessarily isometries, and that this conjecture would imply the preceding conjecture about typical countable dense sets being strongly non-Rado. Its status remains open.