By Sziplrajn's Theorem, we know that every partial order $\succsim$ (i.e. reflexive, transitive and antisymmetric relation) on a nonempty set $X$ can be extended to a linear order (i.e. a complete partial order) on $X$, where an extension of $~\succsim$ is a preorder $~\trianglerighteq$ such that for all $x,y \in X$, $x\succsim y$ implies $x\trianglerighteq y$.
I am looking for a variant of this theorem in which the extension should simply be a preorder (reflexive and transitive), and the initial relation would only be required to be acyclic (an acyclic relation is one for which there exists no list $(x_1, x_2, \dots, x_n)$ with $x_i \in X$ for all $i\in \{1,\dots,n\}$ and $x_1\succsim x_2 \succsim \dots \succsim x_n \succsim x_1$)
So can any acyclic relation be extended to a preorder?