Arnold's characteristic reconstruction conjecture for multidimensional continued fractions

A two-dimensional (or, more generally, $n$-dimensional) continued fraction is represented by a sail, whose facets have homeomorphic types, adjacency relations, integer volumes and integer distances, and whose edges have integer lengths and integer angles between adjacent edges. These characteristics are understood up to the equivalence relation for continued fractions. Characteristic reconstruction conjecture. To construct the whole continued fraction up to equivalence relation it is sufficient to know only the following characteristics for one of the sail: the homeomorphic types of all facets for one of the sails of the fraction, their adjacency to each other; all their integer volumes and integer distances to them; integer lengths of all edges; and all integer angles for any two adjacent edges. The claim addresses the unresolved reconstruction problem for two-dimensional sails, for which no answer was known in the source; the attribution to Arnold is attached to the preceding formulation of the problem rather than explicitly to this conjecture.