The formal adjunction conjecture for weighted cone models

Let $Q$ be a quasi-category tensored and cotensored over $\mathfrak N[\mathcal{K}\!\mathit{an}]$, let $d\colon J\to Q$ be a diagram, and let $p\colon\tilde J\to J$ be a weight. Let $st(p)^{bifib}\colon J\to\mathfrak N[\mathcal{K}\!\mathit{an}]$ be a bifibrant replacement of the adjoint straightening of $p$, and let $[[st(p)^{bifib},d]]$ be the object of $Q$ defined by

$$[[st(p)^{bifib},d]]:=\lim\big(\operatorname{Tw}(J)\to J^{\operatorname{op}}\times J\to\mathfrak N[\mathcal{K}\!\mathit{an}]^{\operatorname{op}}\times Q\to Q\big),$$

using the twisted-arrow projections, $(st(d)^{bifib})^{\operatorname{op}}\times d$, and the cotensor functor. Formal adjunction conjecture. There is a Joyal equivalence

$$st(p)^{bifib}\downarrow_{\mathfrak N[\mathcal{K}\!\mathit{an}]^J}\operatorname{Hom}_{Q}(-,d)\simeq(st(p)^{bifib}\otimes-)\downarrow_{Q^J}d\simeq Q\downarrow_Q[[st(p)^{bifib},d]].$$

The source describes this as likely to be a formal property of two-variable adjunctions and as a way to expand the formula for the weighted cone quasi-category; no proof or resolution is supplied.