$\mathbb{M}\backslash \mathbb{L}$ near $3$

We construct four new elements $3.11>m_1>m_2>m_3>m_4$ of $\mathbb{M}\backslash \mathbb{L}$ lying in distinct connected components of $\mathbb{R}\setminus \mathbb{L}$, where $\mathbb{M}$ is the Markov spectrum and $\mathbb{L}$ is the Lagrange spectrum. These elements are part of a decreasing sequence $(m_k)_{k\in\mathbb{N}}$ of elements in $\mathbb{M}$ converging to $3$ and we give some evidence towards the possibility that $m_k\in \mathbb{M}\setminus \mathbb{L}$ for all $k\geq 1$. In particular, this indicates that $3$ might belong to the closure of $\mathbb{M}\setminus \mathbb{L}$. So, $\mathbb{M}\setminus \mathbb{L}$ would not be closed near $3$ and there would not exist $\varepsilon>0$ such that $\mathbb{M}\cap (-\infty,3+\varepsilon)=\mathbb{L}\cap (-\infty,3+\varepsilon).$