Given a basis in a Banach space $X$, can one find, for every $\varepsilon>0$, an equivalent basis with basis constant at most $1+\varepsilon$?

In $L_p[0,1]$ with $1<p<\infty$ any monotone basis is unconditional so one cannot expect better than $1+\varepsilon$ in general. What about $L_1[0,1]$, is every basis equivalent with a monotone basis?

Edit: The original question has a negative answer, thank you for the reference. What about the $L_p[0,1]$ case? Any known results in this direction?