$M\setminus L$ near 3

We construct four new elements $3.11>m_1>m_2>m_3>m_4$ of $M\mathrm{\backslash }L$ lying in distinct connected components of $\mathbb{R}\setminus L$, where $M$ is the Markov spectrum and $L$ is the Lagrange spectrum. These elements are part of a decreasing sequence $(m_k{)}_{k\in \mathbb{N}}$ of elements in $M$ converging to $3$ and we give some evidence towards the possibility that $m_k\in M\setminus L$ for all $k\ge 1$. In particular, this indicates that $3$ might belong to the closure of $M\setminus L$, so that the answer to Bousch’s question about the closedness of $M\setminus L$ might be negative.