The comparison conjecture for weighted cones and comma quasi-categories

Let $Q$ be a quasi-category, let $d\colon J\to Q$ be a diagram, and let $p\colon\tilde J\to J$ be a weight. Write $st(p)^{bifib}\colon J\to\mathfrak N[\mathcal{K}\!\mathit{an}]$ for a bifibrant replacement of the adjoint straightening of $p$, and let $\operatorname{Hom}_{Q}(-,d)\colon Q\to\mathfrak N[\mathcal{K}\!\mathit{an}]^J$ be the functor of mapping objects into $d$. Comparison conjecture. There is a Joyal equivalence

$$Q^p_{\mathbin{\mkern-1mu{/}\mkern-5mu{/}\mkern-1mu} _d}\simeq st(p)^{bifib}\downarrow_{\mathfrak N[\mathcal{K}\!\mathit{an}]^J}\operatorname{Hom}_{Q}(-,d),$$

where the right-hand side is the comma quasi-category of the functors $st(p)^{bifib}\colon\Delta[0]\to\mathfrak N[\mathcal{K}\!\mathit{an}]^J$ and $\operatorname{Hom}_{Q}(-,d)\colon Q\to\mathfrak N[\mathcal{K}\!\mathit{an}]^J$. This is proposed as a third equivalent model for the quasi-category of weighted cones; the source presents it as an expectation inspired by the ordinary categorical case, and does not state a resolution.