Free-monoid description of iterated category constructions on classifying spaces

Let $M$ be an ungroup-like abelian monoid, let $X$ be a topological space, and define $\mathrm{SP}(X,M)$ to be the free strictly commutative topological monoid generated by $X$-many copies of $M$, characterized by

$$\operatorname{map}_{\mathrm{TopAbMon}}(\mathrm{SP}(X,M),Y)\cong\operatorname{map}_{\mathrm{Top}}(X,\operatorname{map}_{\mathrm{AbMon}}(M,Y))$$

for every abelian topological monoid $Y$. Free-monoid conjecture. There is an equivalence

$$\dot{\mathcal{T}}^n(\mathbf{B}^nM)\simeq\mathbf{B}(\mathrm{SP}(S^{n-1},M)),$$

and the right fibration

$$\dot{\mathcal{L}}^n(\mathbf{B}^nM)\to\dot{\mathcal{T}}^n(\mathbf{B}^nM)$$

corresponds to the action of $\mathrm{SP}(S^{n-1},M)$ on $M$ induced by the topological monoid homomorphism $\mathrm{SP}(S^{n-1},M)\to\mathrm{SP}(*,M)\cong M$. This gives a concrete description of the constructions on the classifying $(\infty,n)$-category $\mathbf{B}^nM$.