Moreira's question on intervals in the Lagrange and Markov spectra

Let $L$ be the classical Lagrange spectrum and $M$ be the classical Markov spectrum. Define

$$d(t)=HD(L\cap (-\infty,t))=HD(M\cap (-\infty,t))$$

and

$$a=\inf\{t\in \mathbb{R}; d(t)=1\}.$$

Moreira's question. For every $\delta>0$, one has

$$\operatorname{Leb}(L\cap (-\infty,a-\delta))=0=\operatorname{Leb}(M\cap (-\infty,a+\delta)),$$

while

$$\operatorname{int}(L\cap (-\infty,a+\delta))\neq \emptyset$$

and

$$\operatorname{int}(M\cap (-\infty,t+\delta))\neq \emptyset.$$

The statement is presented as the authors' hope concerning Moreira's question 1. The supplied text gives no resolution; notably, the final occurrence of $t$ in the Markov-spectrum interval is retained exactly as stated.