The product decomposition conjecture for the Carleman-Sobolev completion

Let $0<p<1$, let $\mathcal{M}=(M_k)_{k\geq 0}$ be the weight sequence defining the quasi-norm

$$\left\lVert f\right\rVert_{\mathcal{M}}=\sup_{k\geq 0}\frac{\left\lVert f\right\rVert_p}{M_k},$$

and let $\widehat{\mathcal{S}}$ be the completion of the subclass $\mathcal{S}$ with respect to this quasi-norm. Assume that $\mu_{\mathcal{M}}=\infty$ and that the regularity assumptions of the cited theorem are satisfied. The product decomposition conjecture. Then

$$\widehat{\mathcal{S}}\cong L^p\times L^p\times L^p\times\cdots.$$

This conjecture concerns the complementary regime in which the smooth embedding result fails: under the stated growth and regularity assumptions, the completion is expected to have the same countable-product structure as the completion of all smooth functions.