A subspace $Y$ of a Banach space is said to be Hahn-Banach smooth if every $f\in Y^*$ has unique norm preserving extension to whole $X$. This notion is related to many other geometric properties of Banach space, viz. rotundity of the dual space, intersection properties of balls in the space etc. It is well known that if $X^*$ is strictly convex then any subspace of $X$ is Hahn-Banach smooth and vice versa-A well-known result by R. R. Phelps. The list also includes the subspaces of type $\{f\in C(K):f|_D=0\}$ in $C(K)$, where $K$ is compact $T_2$ and $D\subseteq K$ is closed. These are so-called M-ideals or the two-sided ideals of a commutative $C^*$ algebra but of course much more stronger than Hahn-Banach smoothness.

My question is if $Y$ is a subspace of a Banach space $X$ which is Hahn-Banach smooth then is it necessary that $Y^{\perp\perp}$ has also similar property in $X^{**}$? One can assume that $X=C(K)$ for some compact Hausdorff $K$. More generally if $X$ is a $L_1$ predual space.