Comparison of precomplicial spaces with Rezk's Θ_n-spaces

Let $n\geq 0$. Write $\mathit{s}\mathcal{S}\!\mathit{et}^{t\Delta^{\operatorname{op}}}$ for the category of prestratified simplicial spaces, equipped with the cartesian model structure whose fibrant objects are called $n$-precomplicial spaces. Let $\mathit{s}\mathcal{S}\!\mathit{et}^{\Theta_n^{\operatorname{op}}}$ denote Rezk's model category of $\Theta_n$-spaces. Comparison conjecture. The model structure on $\mathit{s}\mathcal{S}\!\mathit{et}^{t\Delta^{\operatorname{op}}}$ for $n$-precomplicial spaces is Quillen equivalent to Rezk's model structure on $\mathit{s}\mathcal{S}\!\mathit{et}^{\Theta_n^{\operatorname{op}}}$ for $\Theta_n$-spaces. This would provide the final comparison between the precomplicial-space model and the $\Theta_n$-space model, extending the established comparisons with precomplicial sets and motivating the ongoing work described by the authors.