Symmetric-operator commutant conjecture for the multiqudit Slater construction

Let $P_\mu$ be the local projectors on $\mathbb C^d$, let $\mathcal S$ denote symmetrization over the $d$ tensor factors, and let $\mathfrak A_f$ be the symmetric algebra and $\mathfrak B$ the permutation algebra from the paper's symmetric-algebra construction. For $\mu=0,\dots,N-1$, define

$$T_\mu=d\,\mathcal S(P_\mu\otimes\mathbb I_d\otimes\dots\otimes\mathbb I_d).$$

Equivalently,

$$T_\mu=P_\mu\otimes\mathbb I_d\otimes\dots\otimes\mathbb I_d+\mathbb I_d\otimes P_\mu\otimes\dots\otimes\mathbb I_d+\dots+\mathbb I_d\otimes\dots\otimes\mathbb I_d\otimes P_\mu.$$

Symmetric-operator commutant conjecture. The operators $T_\mu$ generate all the symmetric algebra $\mathfrak A_f$. Equivalently, for

$$\mathcal K=\{T_\mu:\mu=0,\dots,N-1\},$$

one has

$$\mathcal K'=\mathfrak B.$$

That is, the commutant of the set of operators $T_\mu$ is precisely the permutation algebra. The paper presents this as a stronger assertion implying the preceding uniqueness conjecture, and notes that it can be directly verified in specific cases; no general resolution is supplied.