Let $S$ be a (unit) sphere in a Hilbert Space $H$ with $\dim H \ge 3$. Let $A \subset S$ have the following properties:
$A$ is connected;
The affine hull of $A$ is the whole space;
For every $x,y\in A$ there is a unitary operator $T:H\to H$ (or, alternatively, an isometry $T:S\to S$), such that $TA=A$, and such that $Tx=y$.
Is it true that $A=S$?
You can assume that $A$ with the induced strong or weak topology is locally compact and locally linearly connected if needed.
I think that the problem may be meaningful even in dimension $3$, where it would be nice to have an elementary solution.
The problem I have started with is slightly more general, in case of the affirmative answer for this one, there may be a follow-up.