The characterization of 1-connected compact metric spaces by snowflake kernels

Let $4\mathcal{M}^14$ denote the collection of all $1$-connected compact metric spaces, and let $4\mathfrak{S}_p4$ be the operator whose kernel consists of the spaces mapped to the zero space under the $p$-metric construction. Then the characterization conjecture.

$$\mathcal{M}^1=\mathop{\bigcap}_{p>1}\ker(\mathfrak{S}_p).$$

The preceding results establish strict inclusions among the kernels $4\ker(\mathfrak{S}_p)4$ for different values of $p$, but do not determine this intersection. The conjecture proposes that it is exactly the class of all $1$-connected compact metric spaces.