Is there a profinite group $G$ with a binary transitive relation $<$ such that for any different $x,y \in G$ either $x < y$ or $y < x$ and such that for any $x,y,z \in G$ we have that $x < y$ implies that $zx < zy$ ?

Is there a nontrivial profinite group $G$ with a binary transitive relation $<$ such that

1. $x<y$ implies $x\neq y$, and for any different $x,y \in G$ either $x < y$ or $y < x$ and such that for any $x,y,z \in G$ we have that $x < y$ implies that $zx < zy$ (i.e., $<$ defines a left-invariant strict total order)
2. $\{(x,y):x<y\}$ is open?