The Mathieu xz-conjecture for moments of admissible functions

Let $g$ and $h$ be admissible functions on $\mathbb{T}^k\times[0,1]^l$, with admissibility as in the xz-conjecture. Mathieu xz-conjecture. If

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

for every positive integer $P$, then

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

for all sufficiently large $P$. The source states that this consequence follows from the xz-conjecture; no independent resolution is given.