I have not yet been able to find a direct wave-mechanical explanation (which would bypass the above semiclassical estimate) for the quasi-orthogonality of dilation. However a clue is given by Berry and Wilkinson's proof  that degenerate () off-diagonal elements are exactly zero. An explanation for was first attempted by VS [195,194]. They tried to establish quasi-orthogonality using the identity , with . This identity can be proved in a simple fashion (Eq.(H.25)). However, they then made the assumption (for ) by claiming that eigenstates are uncorrelated across the volume of . (In this would give variables whose variances add linearly). This would give a power law for the off-diagonal growth of .
This is in error for two reasons. Firstly, we know that the number of degrees of freedom in a constant-wavenumber function is actually scaling like the boundary (see Section 5.3.1), which would imply the naive random-wave estimate . Secondly, a random wave estimate of an overlap over a large fraction of the region is generally very bad (Section 3.3.2). In fact, comparison to our band profile results (analytical and numerical) shows that , a result which cannot be guessed by random wave assumptions. This result has also since been verified by Vergini 6.1.
The conclusion is that the original authors' quasi-orthogonality estimate (which they have used in [195,194,175,196,203]) was in fact pessimistic: the true power law actually gives much smaller elements close to the diagonal.