The unit-group 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 of real representations, and let $U(\mathcal{R}(S_n))$ be the group of units of this ring. Unit-group conjecture. For any $n\in\mathbb{N}$, the unit group $U(\mathcal{R}(S_n))$ is isomorphic to the Klein four-group.

The result has been checked in low dimensions, but the existing computation depends on explicit tables of irreducible characters and does not extend directly to general $n$; an alternative proof is sought.