Sobolev Regularity of Convex Envelope

Question

Suppose I know the function $f \in W^{2,p}$ for some $p>p*$ where p* is the Sobolev exponent needed in order for f to embed into $C^{1,\alpha}$. Then classical regularity theory for convex envelopes tells you that $f^{**}$ also embeds into the same $C^{1,\alpha}$ but can we also say it embeds into the same Sobolev Space? I can't seem to find any theorems along these lines. Alexandroff gives you that a second weak derivative exists but I need some high Lp bounds on it.