two-sided sgut-majorization and its linear preservers

Let $textbf{m}_{n,m}$ be the set of all $n$-by-$m$ real matrices, and let $mathbb{r}^{n}$ be  the set of all $n$-by-$1$ real vectors. an $n$-by-$m$ matrix $r=[r_{ij}]$ is called g-row substochastic if $sum_{k=1}^{m} r_{ik}leq 1$  for all $i     (1leq i leq n)$.  for $x$, $y in mathbb{r}^{n}$, it is said that $x$ is $textit{sgut-majorized}$ by $y$, and we write  $ x    prec_{sgut}y$  if there exists an $n$-by-$n$ upper triangular g-row substochastic matrix $r$ such that $x=ry$. define the relation $sim_{sgut}$ as follows. $xsim_{sgut}y$ if and only if $x$ is   sgut-majorized  by $y$ and $y$ is sgut-majorized  by $x$.  this paper characterizes all (strong)  linear preservers of  $sim_{sgut}$ on $mathbb{r}^{n}$.