Strong mixing and uniform density conjecture for LLL coefficients

Let $\mathcal{D}$ be a “generic” distribution on the set of bases in $\mathbb{R}^n$ used to sample inputs for LLL. For the pile toppled at iteration $i$, let $k(i)$ denote its index, so that $\mu_{k(i)}$ is a random variable determined by the input distribution.

LLL mixing and density conjecture. (i) The sequence $(|\mu_{k(i)}|)_{i=1,2,\ldots}$ is strongly mixing as a stochastic process. (ii) Each $|\mu_{k(i)}|$ is contained in a compact subset $S$ of the set of all probability density functions on $[0,0.5]$ with respect to the $L^\infty$-norm, where $S$ is independent of the dimension, the input distribution, and any other variable.

This is proposed as a rigorous version of the claim that LLL is essentially a sandpile model. The first part formalizes the heuristic that widely separated coefficients are nearly independent; the second asserts a dimension- and distribution-independent compactness property. The source gives no resolution of either assertion.