The choice of $\phi$ affects:
Time-shift invariance: slower decay in time will increase it
Time-warp stability: slower decay in time will increase it mainly for deformations along time (but not only time; Ctrl +F "frequential averaging effect" here)
Time-frequency resolution: anything other than Gaussian (or DPSS) will yield a suboptimal joint ...
It's indeed only a statement on proportionality. Wavelets aren't compat, so any notion of "support" invokes a heuristic (engineered criterion).
However, it's not applicable to all wavelets: Morlets are unimodal and symmetric in time and frequency, which enables inverse proportionality (counterexample).
Kymatio defines "support" as ...
It keeps the scale of the largest scale wavelet $\leq T$ while still tiling the entire frequency axis.
using a single low pass filter built with a single scaling function
would not achieve this: $\phi$ of scale $T$, and $\psi$ whose temporal widths don't taper, will result either in $\psi$ whose width is $>T$, or in incomplete tiling.
Note that a jump ...