Decay preservation for quotient spaces of integral-operator algebras

Let $\mathcal L^1_v\circ$ and $\mathcal L_v\circ$ be extensions of $\mathcal L^1_v$ and $\mathcal L_v$ that also contain non-invertible operators, and let $K_{\mathcal L^1_v\circ}$ and $K_{\mathcal L_v\circ}$ be their respective kernels. Define

$$\mathcal L^{1'}_v=\mathcal L^1_v\circ/K_{\mathcal L^1_v\circ},\qquad \mathcal L'_v=\mathcal L_v\circ/K_{\mathcal L_v\circ}.$$

Quotient-space decay conjecture. The spaces $\mathcal L^{1'}_v$ and $\mathcal L'_v$ possess the same decay properties as $\mathcal L^1_v$ and $\mathcal L_v$, respectively. The source presents this as a proposed extension of known results to quotient spaces and supplies no proof or resolution.