2 problems

Refined flat-facet conjecture. The event that there are such planes for which $L_i\cap\Gamma(\sigma)$ are flat facets in the following sense has probability tending to $1$ as…

Faceted-droplet hypothesis. As $l\to\infty$, with probability tending to $1$, there exist six distinct two-dimensional planes $L_i=L_i(\sigma)\subset\mathbb{T}^3$, $i=1,\ldots,6$,…