Partitioning the set of Pauli words into abelian pieces

Let $$\sigma_x,\sigma_y,\sigma_z$$ be the Pauli matrices. A Pauli word of length $$n$$ is defined as the tensor product $$\otimes_{i=1}^n\sigma_i$$ of operators $$\sigma_1,\dots,\sigma_n\in\{\mathbf 1,\sigma_x,\sigma_y,\sigma_z\}$$. The set $$PW_n$$ of Pauli words of length $$n$$ has cardinality $$4^n$$. A subset $$A\subset PW_n$$ is called abelian if $$u\circ v=v\circ u$$ for any $$u,v\in A$$.

Question. What is the number of partitions of the set $$PW_n\setminus \{\otimes_{i=1}^n\mathbf 1\}$$ into $$2^n+1$$ abelian subsets $$A_{1},\dots,A_{2^n+1}$$ of cardinality $$|A_i|=2^n-1$$?

This problem was posed in December 2019 by a Francis Drake (probably, a pseudo) on page 41 of Volume 3 of the Lviv Scottish Book.

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