Weakly unital strict groupoids model truncated spaces

Let $n$ be a nonnegative integer. Consider the category of weakly unital strict $n$-groupoids, its equivalences, and the category of $n$-truncated spaces with weak equivalences. Let $\tau_{}$ denote localization at the indicated class of morphisms. Weakly unital strict groupoid conjecture. There are functors

$$\Pi_n$$

and

$$\Re$$

between weakly unital strict $n$-groupoids and $n$-truncated spaces, in the usual opposite directions, together with adjunction morphisms inducing an equivalence

$$\tau_{\mathrm{equivalences}}(\mathrm{weakly\ unital\ strict\ }n\text{-}\mathrm{groupoids})\simeq\tau_{\mathrm{weak\ equivalences}}(n\text{-}\mathrm{truncated\ spaces}).$$

The source reports that Joyal and Kock proved this conjecture for $n=3$; the general case is left unresolved there.