Nonexistence of a continuum-separable envelope for infinite-dimensional Lebesgue measure

Let $\mu$ be the infinite-dimensional Lebesgue measure on $\mathbb{R}^{\mathbb{N}}$, and let $\mathfrak{c}$ denote the cardinality of the continuum. A $\mathfrak{c}$-separable envelope is the relevant envelope associated with the measure space $(\mathbb{R}^{\mathbb{N}},\mu)$ whose separability cardinal is at most $\mathfrak{c}$. Envelope conjecture. There is a model of ZFC in which $(\mathbb{R}^{\mathbb{N}},\mu)$ does not admit a $\mathfrak{c}$-separable envelope. The preceding discussion shows that the case $p=2$ has a straightforward Hilbert-space classification, while the conjecture concerns the measure-theoretic envelope structure of the infinite-dimensional Lebesgue space and its dependence on the set-theoretic model.