The xz-conjecture for Laurent polynomials with polynomial coefficients

Let $h$ be an admissible function on $\mathbb{T}^k\times[0,1]^l$ of the form

$$h(z_1,\ldots,z_k;x_1,\ldots,x_l)=\sum_{\mathbf m}c_{\mathbf m}(x)z^{\mathbf m},$$

where $\mathbf m=(m_1,\ldots,m_k)$ is an integer multi-index, each $c_{\mathbf m}$ is a complex polynomial in the $x_j$, and $\operatorname{Sp}(h)$ is the set of $\mathbf m$ for which $c_{\mathbf m}\neq0$. The xz-conjecture. If

$$\int_{[0,1]^l}\int_{\mathbb{T}^k}h^P=0$$

for every positive integer $P$, then the zero vector is not in the convex hull of $\operatorname{Sp}(h)$. This is known when there are no $x$-variables and when there is one $x$-variable and no $z$-variables, but remains open already for one $x$ and one $z$.