The Harrison-centre conjecture for representation rings of symmetric groups

Let $S_n$ be the symmetric group on $n$ letters, let $\mathcal{R}(S_n)$ denote its representation ring, and let $f_{S_n}$ be the relevant element defining the Harrison centre $Z(f_{S_n})$. Harrison-centre conjecture. For any $n\in\mathbb{N}$, the representation ring satisfies

$$\mathcal{R}(S_n)\cong Z(f_{S_n}).$$

The paper proves this isomorphism for $S_5$ and $S_6$, while the general case is posed as a remaining problem because the available method relies on explicit irreducible-character data.