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.