Weighted decay conjecture for integral operators

Let $h\in L^1(\mathbb R^d)$, and let $h_L$ denote the associated kernel or integral operator in the notation of the source. For $\delta>0$, define

$$\tau_\delta(x)=(1+|x|)^\delta.$$

Weighted decay conjecture. There exists $\delta>0$ such that $h_L\in\mathcal L^1_{\tau_\delta}$. If true, this would support identifying $\mathcal L^1$ with a union of the weighted spaces and could imply symmetry of the Banach algebra, subject to the additional norm and lemma conditions described in the source.