  We construct four new elements $3.11>m_1>m_2>m_3>m_4$ of $M\backslash L$
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 $k\geq 1$. 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.
