Weak real zero amalgamation conjecture for polynomials on two-element overlaps

Let $E'=E_1'\cup E_2'$ be a finite set, and let $E_0'=E_1'\cap E_2'$ have two elements. For $k=1,2$, let $Q_k\in\mathbb{R}[x_i\mid i\in E_k']$ be real zero polynomials. Suppose that

$$Q_1|_{x_i=0\,\text{for }i\notin E_0'}=Q_2|_{x_i=0\,\text{for }i\notin E_0'}.$$

Weak real zero amalgamation conjecture. There should be a real zero polynomial $Q\in\mathbb{R}[x_i\mid i\in E']$ such that, for $k=1,2$,

$$Q_k=Q|_{x_i=0\,\text{for }i\notin E_k'}.$$

This is presented as a more specific formulation of the real zero amalgamation conjecture, which the paper then disproves.