Baez–Breen–Dolan stabilization hypothesis for weak n-categories

Let $n\ge 0$ be fixed. For $k\ge 0$, define recursively the dendroidal set $wCat_k^n$ of weak $k$-monoidal $n$-categories by

$$wCat_0^n=\,{}_nCat,$$

and, for $k>0$,

$$wCat_k^n=[A,wCat_{k-1}^n],\qquad A=N_d(As).$$

A dendrex of shape $\eta$ in $wCat_k^n$ is called a $k$-monoidal $n$-category. Baez–Breen–Dolan stabilization hypothesis. For a fixed $n\ge 0$, there is an isomorphism of dendroidal sets

$$wCat_k^n\cong wCat_{n+2}^n$$

for any $k\ge n+2$. This is the stabilization prediction for the hierarchy of weak monoidal structures: beyond level $n+2$, increasing the monoidal degree should add no new information. The statement is made here for the specific dendroidal-set model of weak $n$-categories and is presented as a conjectural hypothesis.