Let $p:X^{\natural} \to S$, $q:Y^{\natural} \to S$ be two cartesian fibrations of simplicial sets (where the marking is given by cartesian edges) and assume that we are given an equivalence of cartesian fibrations $X^{\natural} \to Y^{\natural}$.

Given a marking on $S$ we define the marked simplicial set $X^{\dagger}$ (resp. $Y^{\dagger}$) by declaring and edge to be marked if and only if it is marked in $X^{\natural}$ and its image under $p$ is marked in $S$.

My question is the following: Is the induced map $X^{\dagger} \to Y^{\dagger}$ a weak equivalence of marked simplicial sets (over the point)?