Hamilton-cycle profile conjecture for the hypercube

Let $Q^n$ be the $n$-dimensional hypercube, and let the profile of a subgraph be the tuple recording the number of its edges in each coordinate direction. A tuple is even if every coordinate is even. Hamilton-cycle profile conjecture. Every even tuple $\bm{x}\in\mathbb{N}^n$ satisfying

$$||\bm{x}||_1=2^n,$$ $$\max_i\{x_i\}\leq 2^{n-1},\qquad \min_i\{x_i\}\geq 2$$

is the profile of a Hamilton cycle in $Q^n$. These conditions are necessary: a Hamilton cycle has $2^n$ edges, uses each direction at most $2^{n-1}$ times and at least twice, and its profile is the sum of two perfect matching profiles. The conjecture asserts that these are the only restrictions.