Asymptotic prevalence of lex-segment exterior shiftings

Let $X$ be a fixed compact connected surface. Let $s(X,n)$ be the number of triangulations $T$ of $X$ on $n$ labeled vertices, and let $l(X,n)$ be the number of such triangulations for which $\Delta(T)_1$ is a lex-segment and $\Delta(T)_2\setminus\{2,3,4\}$ is a lex-segment, where $\Delta(T)$ denotes the exterior algebraic shifting.

Lex-segment prevalence conjecture. As $n$ tends to infinity,

$$\frac{l(X,n)}{s(X,n)}\longrightarrow 1.$$

This predicts that, for every fixed compact connected surface, almost all labeled triangulations have the specified lex-segment form after exterior shifting. The statement appears as a direction for further research, and the source gives no resolution.