Reality conjecture for the roots of the polynomial S in Cayley conditions

Let

$$R(x)=x\prod_{i=1}^{2n-1}(x-\gamma_i),$$

where $0<\gamma_{2n-1}<\cdots<\gamma_1$, and let $S(x),P(x)\in\mathbb{R}[x]$ satisfy the relation

$$S(x)^2R(x)=P(x)(P(x)-P(0)),$$

with $P(0)\neq 0$. Reality conjecture for $S$. The polynomial $S(x)$ has only real roots.

The claim concerns the expected real-rootedness of the auxiliary polynomial arising in the algebraic formulation of Cayley conditions. The source gives theoretical evidence and proves the conclusion when $m\leq n+3$, but leaves the general assertion unproved.