The following open problem is taken from the book Open Problems in the Geometry and Analysis of Banach Spaces, page $40.$
Problem $84:$ Assume that $X$ is an infinite-dimensional separable Banach space such that for any pair of points $x$ and $y$ in the unit sphere of $X$ there exists a linear isometry from $X$ onto $X$ such that $Tx=y.$ Is $X$ linearly isometric to a Hilbert space?
After stating the problem, authors mentioned that the problem holds true for finite-dimensional spaces but not for non separable spaces. Then they point to the reference,page $255$ which contains the problem above.
My question: Does anyone know whether the problem is still open?If yes, what is a good reference?
The authors of the book mentioned that problem $84$ is an old classical open problem. So I suspect that there should be some papers available on solving the problem partially or fully.