If you buffer your filters, it seems to work. See attached screenshots.
Each screenshot shows one pot (LPF, MID, HPF) in the max. position, the rest on min.
The sweep is from 100Hz to 15kHz and back; upper showing the source, lower showing the output (100M load).
The ALL.png shows all pots to the max which produces a bit of a lumpy output; not flat.
Actually, given the Vrms differences in the outputs shown, you're probably going to have to trim/scale each stage for a normalized output.
Play with it
here; see right margin for LPF,HPF,MID pot sliders, max is to the left, min to the right.