Strong real zero amalgamation conjecture

Let $\ell=2$, i.e., $x=(x_1,x_2)$, and let $d\in\mathbb N_0$. Suppose $p\in\mathbb R[x,y]$ and $q\in\mathbb R[x,z]$ are real zero polynomials of degree at most $d$ such that $p(x,0)=q(x,0)$. Strong real zero amalgamation conjecture. There exists a real zero polynomial $r\in\mathbb R[x,y,z]$ of degree at most $d$ such that

$$p=r(x,y,0)\qquad\text{and}\qquad q=r(x,0,z).$$

This degree-preserving amalgamation property would strengthen the preceding weak real zero amalgamation conjecture and is motivated by degree-preserving approaches to real amalgamation. Its status is not resolved in the supplied source.