Stabilization conjecture for the spaces ${}_{\Sigma}L_p^k$

For fixed $d$, $p$, and polynomial $\mathfrak{S}$, define

$${}_{\Sigma}L_p^k=\operatorname{clos}_{L_p}\left(\left\{\mathfrak{S}^k\left(\frac{\partial}{2\pi i}\right)[g]\mid g\in\mathcal{S}(\mathbb{R}^d)\right\}\right).$$

Stabilization conjecture. There exists $l$ such that

$${}_{\Sigma}L_p^l={}_{\Sigma}L_p^{l+1}={}_{\Sigma}L_p^{l+2}=\cdots.$$

Thus, the chain of these spaces should stabilize. The source provides no resolution.