There are two ways to stabilize a Legendrian knot $k$ in standard contact sphere $(S^3,\xi_{st})$ i.e. adding right cusps or left cusps, let's call these two stabilized Legendrian knots $k_R$ and $k_{L}$ respectively. If the knot $k$ is embedded on the page of an open book $(\Sigma,\phi)$ of $(S^3,\xi_{st})$, then $k_R$ and $k_L$ can also be embedded on page of an open book obtained by positive stabilization of $(\Sigma,\phi)$. I want to know what are the monodromies of the two new open books relative to $(\Sigma,\phi)$.

I know they cannot be equivalent, since $(-1)$-contact surgery on each of the stabilized knots gives two non-homotopic contact structures (their first chern class is different), so the contact structures are not isotopic and therefore the open books cannot be equivalent.