Near-triviality conjecture for isometries of complex HI spaces

Let $X$ be a complex HI space, meaning a hereditarily indecomposable Banach space, and let the group of isometries of $X$ act on $X$. Near-triviality conjecture. The group of isometries acts nearly trivially on $X$.

This would extend the paper's result for a particular complex HI space without assuming reflexivity or non-existence of a Schauder basis; the source presents it as a conjecture, and no resolution is supplied.