Why are wavelet transforms not scale-equivariant?

One can rely on continuous wavelets to build a multi-resolution analysis that is equivariant (covariant) under the action of a discrete subgroup of translation.

When not downsampled, the multi-resolution analysis of a 1D signal can be seen as a matrix of n x m coefficients, where n are the octaves that one wants to capture, and m are the number of considered translated wavelets on each octave.

Equivariance to translation in this case means that a certain translation of the input leads to a translation (permutation) in the output, thus the response of the action of such a translation gets linearized by a permutation matrix.

I understand this linear commutation is a left regular representation of the translation subgroup.

Why is such a MRA not scale-equivariant, too? Why one would not think that the multiresolution would also respond linearly to dilated versions of the same input? Would a logarithmic resampling of the input help?