Support containment for measures associated with nested polynomial sets

Let $A$ and $B$ be finite sets of integer polynomials, each polynomial having only real roots. For each such set, let $\eta_A$ denote the lower-bound measure and let $\mu_A$ denote the measure produced by the paper's method; write $\operatorname{supp}(\mu)$ for the support of a measure.

Support-containment conjecture. If $A\subset B$, then

$$\operatorname{supp}(\eta_B)\subset\operatorname{supp}(\eta_A)$$

and

$$\operatorname{supp}(\mu_B)\subset\operatorname{supp}(\mu_A)\subset\operatorname{supp}(\eta_A).$$

The containments have been observed in the cases computed in the paper and in the cited prior work, but no general proof is given.