It was proved in [Bennett, G.; Dor, L.E.; Goodman, V.; Johnson, W.B.; Newman, C.M. On uncomplemented subspaces of $L_p$, $1<p<2$. Israel J. Math. 26 (1977), 178–187] that, for $1 <p < 2$, there is an uncomplemented subspace of $L_p(0,1)$ that is isomorphic to $\ell_2$.

Moreover it is well-known that if $1<p<q<2$ and $M$ is an uncomplemented copy of $\ell_2$ in $L_q(0,1)$, then $M$ is also an uncomplemented copy of $\ell_2$ in $L_p(0,1)$.

**My question**: Is it possible to find a subspace $M$ of $L_2(0,1)$ such that, for some $\varepsilon>0$, $M$ is an uncomplemented copy of $\ell_2$ in $L_p(0,1)$ for $2-\varepsilon <p<2$ (hence for $1\leq p<2$)?.