Binary-composition conjecture for strongly regular slices of contractible operads

Let $A$) be a contractible $n$-operad containing a system of binary compositions, and suppose that all its slices up to dimension $n-1$ are strongly regular theories. A weak $n$-category is an algebra equipped with the structure governed by a contractible $n$-operad.

Binary-composition conjecture. Every weak $n$-category is weakly equivalent to an $A$-algebra.

The conjecture proposes that strong regularity of the lower-dimensional slices is the appropriate analogue of freeness of the symmetric-group actions for controlling the homotopy behaviour of higher operads. The supplied text gives examples motivating the condition but no resolution.