Equivalent definitions of Cartesian Fibrations between Quasi-Categories

Riehl and Verity prove that their definition agrees with Lurie's in Corollary 4.1.24 (cf. Remark 2.4.1.4 in HTT).

Here's an outline of a proof that Riehl-Verity (RV) cartesian fibrations are the same as Joyal-Lurie (JL) cartesian fibrations. There are some details that I'm not sure actually hold though.

First show that $p: E \to B$ is a JL cartesian fibration iff the cotensor $p^{\Delta^n} : E^{\Delta^n} \to B^{\Delta^n}$ is an RV cartesian fibration for every $n$. In fact, I think that a lift is JL-cartesian iff its degeneracies in $E^{\Delta^n}$ are RV-cartesian for every $n$.

Then use Riehl and Verity's Theorem 4.1.10, which says that an isofibration $p: E \to B$ is an RV cartesian fibration iff the induced functor $E\downarrow E \to B \downarrow p$ admits a right adjoint right inverse. Now cotensoring with $\Delta^n$ induces should commute with taking comma categories, and I think (or hope) that it should be 2-functorial on the homotopy 2-category. So if $E\downarrow E \to B \downarrow p$ admits a right adjoint right inverse, then $E^{\Delta^n} \downarrow E^{\Delta^n} \to B^{\Delta^n} \downarrow p^{\Delta^n}$ does too, and so $p^{\Delta^n}: E^{\Delta^n} \to B^{\Delta^n}$ is RV-cartesian, and $p$ is JL cartesian.