Iterated lax monoid bicategory conjecture

Let $\mathcal{B}$ be a monoidal bicategory. For each monoid object in $\mathcal{B}$, consider lax monoidal $1$-cells and monoidal $2$-cells. Iterated lax monoid conjecture. It should be possible to define a monoidal bicategory $\operatorname{Lax}(\mathcal{B})$ of monoid objects, lax monoidal $1$-cells, and monoidal $2$-cells in $\mathcal{B}$, such that

$$\operatorname{Lax}(\operatorname{Lax}(\mathcal{B}))$$

is the monoidal bicategory of lax double monoid objects,

$$\operatorname{Lax}(\operatorname{Lax}(\operatorname{Lax}(\mathcal{B})))$$

is the monoidal bicategory of lax triple monoid objects, and so on. Consequently,

$$\operatorname{Lax}^n(\operatorname{Cat})$$

would be the monoidal bicategory of $n$-fold monoidal categories, fully lax monoidal functors, and monoidal natural transformations.

This conjecture would organize lax double, triple, and higher monoid objects through iteration of a single monoidal-bicategorical construction. The source gives no evidence that the construction or its asserted identifications have been established.