Definite symmetric determinantal representation from Rayleigh differences

Let $f\in\mathbb{R}[x_1,\ldots,x_m]$ be multiaffine in $x_1,\ldots,x_n$, with nonzero coefficient of $x_1\cdots x_n$. For indices $i,j$, let $\Delta_{ij}(f)$ denote the corresponding Rayleigh difference. Definite symmetric determinantal representation conjecture. Some power of $f$ has a definite real symmetric determinantal representation if and only if, for all $i,j$, $\Delta_{ij}(f)$ is a sum of squares in $\mathbb{R}[x_1,\ldots,x_m]$. This is presented as a hoped-for partial converse to a known result relating definite determinantal representations to sums of squares; its resolution is not given in the source.