Dimension conjecture for quadratic motherbody equations

Fix a monic polynomial $P(z)$ of degree $n+2$ and a polynomial $Q(z)$ of degree at most $n+1$. Let the quadratic equation referred to in the source as $(\mathrm{quadr})$ be the equation under consideration, and define $\Omega_{P,Q}$ to be the set of polynomials $R(z)$ of degree at most $n$ for which this equation admits a probability measure, meaning a positive motherbody measure of mass $1$.

Dimension conjecture for quadratic motherbody equations. The set $\Omega_{P,Q}$ is a real semi-analytic variety of real dimension $n$.

The conjecture seeks a general description of the parameter set for quadratic equations admitting probability motherbody measures. The source says that rigorous results about the structure of this set were not then available, while presenting the stated dimension as a proposed consequence of generalizing earlier methods.