Schweighofer–Sawall weak real zero amalgamation conjecture

A real zero polynomial is a polynomial $P\in\mathbb{R}[x_1,\ldots,x_n]$ with $P(0)\neq 0$ whose restriction to every real line through the origin has only real zeros. Let $F\in\mathbb{R}[x_1,x_2,y_1,\ldots,y_m]$ and $G\in\mathbb{R}[x_1,x_2,z_1,\ldots,z_n]$ be real zero polynomials. If

$$F|_{y_1=\cdots=y_m=0}=G|_{z_1=\cdots=z_n=0},$$

Schweighofer–Sawall's weak real zero amalgamation conjecture. There should be a real zero polynomial $H\in\mathbb{R}[x_1,x_2,y_1,\ldots,y_m,z_1,\ldots,z_n]$ such that

$$F=H|_{z_1=\cdots=z_n=0}\quad\text{and}\quad G=H|_{y_1=\cdots=y_m=0}.$$

The paper provides a counterexample, so the conjecture is false.