  The two most fundamental conjectures on the structure of the generic Hecke
algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state
that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of
definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace.
The first conjecture has recently become a theorem, while the second
conjecture, known to hold for real reflection groups, has only been proved for
some exceptional non-real complex reflection groups (all of rank $2$ but one).
The two most fundamental conjectures on the structure of the parabolic Hecke
subalgebra $\mathcal{H}(W')$ associated with a parabolic subgroup $W'$ of $W$
state that $\mathcal{H}(W)$ is a free left and right $\mathcal{H}(W')$-module
of rank $|W|/|W'|$, and that the canonical symmetrising trace of
$\mathcal{H}(W')$ is the restriction of the canonical symmetrising trace of
$\mathcal{H}(W)$ to $\mathcal{H}(W')$. Until now, these two conjectures have
only be known to be true for real reflection groups. We prove them for all
complex reflection groups of rank $2$ for which the BMM symmetrising trace
conjecture is known to hold.
