This is a follow up to the questions posed in Generalization of the Hermite-Bielher-Kakeya Theorem. Here is an interesting follow up to those comments.
Firstly we remark that: $f(x)+g(x)\cdot w$ is stable for all $w\in\mathbb{C}$ iff $f(x)+g(x)\cdot r$ is hyperbolic for all $r\in\mathbb{R}$. Hyperbolic means that a polynomial has only real zeros.
If either the "stability" condition holds or the "hyperbolic" condition holds then $f$ and $g$ must have interlacing zeros. Understanding this fact, we pose a more subtle extension (than the generalized statement from above). Can we show:
Theorem. Suppose $f_n$, $f_{n-1}$, and $f_{n-2}$ are real-valued polynomials with degrees, $n$, $n-1$, and $n-2$ respectively, with positive leading coefficients. Suppose further that for every $r\in\mathbb{R}$, $$f_n(x)r^2+f_{n-1}(x)r+f_{n-2}(x)$$ is hyperbolic. Then $f_n$, $f_{n-1}$, and $f_{n-2}$ have only real zeros and moreover $f_n$ interlaces $f_{n-1}$ and $f_{n-1}$ interlaces $f_{n-2}$.
We fist note, by taking limits that certainly $f_n$ and $f_{n-2}$ have only real zeros. But how would we prove that $f_{n-1}$ has only real zeros? Is there a counterexample? How about the interlacing condition. In some sense it seems that condition $$f_n(x) r^2+f_{n-1}(x)r+f_{n-2}(x)\in\text{Hyperbolic}$$ is stronger than the two conditions $$f_n(x)r+f_{n-1}(x)\in\text{Hyperbolic}\ \ \ \ \ \ \ \ \ f_{n-1}(x)r+f_{n-2}(x)\in\text{Hyperbolic}.$$ If this implication of conditions is true, then the theorem would follow.
