The product decomposition conjecture for the completion of smooth functions

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 $\widehat{\mathcal{C}}$ be the completion of $C^{\infty}$ with respect to this quasi-norm. The product decomposition conjecture. For the space $\widehat{\mathcal{C}}$ we have

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

This conjecture proposes that the completion of the whole smooth-function space decomposes topologically as a countable product of copies of $L^p$, extending the pathological behavior known for finite-order Sobolev spaces when $0<p<1$.