Coherence conjecture for n-fold monoidal categories

Let $\mathcal{C}$ be an $n$-fold monoidal category with products $\otimes^{(1)},\ldots,\otimes^{(n)}$, units, associators, unitors, and interchange morphisms. For families of objects $X_{a_1,\ldots,a_n}$ indexed by sets $A_1,\ldots,A_n$, consider maps

$$\bigotimes^{(1)}_{a_1\in A_1}\cdots\bigotimes^{(n)}_{a_n\in A_n}X_{a_1,\ldots,a_n}\longrightarrow\bigotimes^{(n)}_{a_n\in A_n}\cdots\bigotimes^{(1)}_{a_1\in A_1}X_{a_1,\ldots,a_n}.$$

Coherence conjecture. All such maps defined as composites of associators, unitors, and interchange morphisms are equal.

This would extend Mac Lane's coherence theorem from monoidal categories to the multiple interacting monoidal structures considered here. The source presents it as a conjectural coherence result, and gives no resolution.