Two sided Sglt-Majorization on R2 and its Linear Preservers

A matrix $R$ is said to be $\textit{g-row substochastic}$ if $Re\leq e$. For $x$, $y \in \mathbb{R}^{n}$, it is said that $x$ is $\textit{sglt-majorized}$ by $y$, $x \prec_{sglt} y$, if there exists an $n$-by-$n$ lower triangular g-row substochastic matrix $R$ such that $x=Ry$. We define two sided sglt-majorization as follows: $ x \sim_{sglt} y$ if $ x \prec_{sglt} y \prec_{sglt} x$. We wish to find the structure of all linear preservers of $\sim_{sglt}$ on $\mathbb{R}^2$.