  E. Oja, T. Viil, and D. Werner showed, in [Totally smooth renormings, Archiv
der Mathematik, 112, 3, (2019), 269--281] that a weakly compactly generated
Banach space $(X,\|\cdot \|)$ with the property that every linear functional on
$X$ has a unique Hahn--Banach extension to the bidual $X^{**}$ (the so-called
Phelps' property U in $X^{**}$, also known as the Hahn--Banach smoothness
property) can be renormed to have the stronger property that for every subspace
$Y$ of $X$, every linear functional on $Y$ has a unique Hahn--Banach extension
to $X^{**}$ (the so-called total smoothness property of the space). We mention
here that this result holds in full generality -- without any restriction on
the space -- and in a stronger form, thanks to a result of M. Raja, [On dual
locally uniformly rotund norms, Israel Journal of Mathematics 129 (2002),
77--91].
