The weak equivalence conjecture for the nerve of unbiased bicategories

Let $ubi\mathcal{C}tg$ be the category of classical bicategories with strict unit, let $N(ubi\mathcal{C}tg)$ denote its nerve, and let $i^*(w\mathcal{C}at^2)$ be the simplicial set associated to dendroidal weak $2$-categories. The inclusion is a map of simplicial sets

$$N(ubi\mathcal{C}tg)\longrightarrow i^*(w\mathcal{C}at^2).$$

Weak equivalence conjecture. The inclusion of simplicial sets $N(ubi\mathcal{C}tg)\longrightarrow i^*(w\mathcal{C}at^2)$ is a weak equivalence in the Joyal model structure on $s\mathcal{S}ets$.

This would strengthen the preceding result that the homotopy category of $i^*(w\mathcal{C}at^2)$ is isomorphic to the category of classical bicategories, by asserting an equivalence before passing to the homotopy category. The source presents this as a conjecture; no resolution is given here.