  We provide an order of convergence for a version of the Carath\'eodory
convergence for the multiple SLE model with a Dyson Brownian motion driver
towards its hydrodynamic limit, for $\beta=1$ and $\beta=2$. The result is
obtained by combining techniques from the field of Schramm-Loewner Evolutions
with modern techniques from random matrices. Our approach shows how one can
apply modern tools used in the proof of universality in random matrix theory,
in the field of Schramm-Loewner Evolutions.
