The totally stable matrix conjecture

Let $f(x,y)=\sum f_i(x)y^i\in\mathbf{P}^{\mathrm{pos}}_2$, and form the upper-triangular coefficient matrix

$$\varphi(f)(x,y)= \begin{pmatrix} f_0&f_1&f_2&\cdots\\ 0&f_0&f_1&\cdots\\ 0&0&f_0&\cdots\\ \vdots&\vdots&\vdots&\ddots \end{pmatrix}.$$

A minor means the determinant of any finite square submatrix. The totally stable matrix conjecture. Every minor of $\varphi(f)(x,y)$ is a stable polynomial. The source describes this as being suggested by empirical evidence and does not prove it in general.