Asymptotic conjectures for the crossroads sequences $L_n$ and $M_n$

Let $L_n$ and $M_n$ be the two integer sequences studied in the paper, and let $C_n$ denote the $n$th Catalan number. The known values include $M_n>L_n$ for $9\leq n\leq14$.

Conjectures for $L_n$ and $M_n$. The following five propositions are conjectured:

$$\forall n\in\mathbb{N},\ n\geq 9\Rightarrow M_n>L_n,$$ $$\lim_{n\to+\infty}\frac{M_n}{L_n}=+\infty,$$ $$\lim_{n\to+\infty}\frac{M_n}{C_n}=1,$$ $$\lim_{n\to+\infty}\frac{L_n}{C_n}=0,$$ $$\lim_{n\to+\infty}\frac{M_{n+1}}{M_n}=\lim_{n\to+\infty}\frac{L_{n+1}}{L_n}=4.$$

The sequences $L_n$ and $M_n$ have only been computed for small values because the available algorithms have exponential complexity. These conjectures describe their eventual ordering, their asymptotic comparison with each other and with the Catalan numbers, and their limiting consecutive-term growth.