You can use opamps to synthesize a negative capacitance, but this is merely equivalent to a differentiator, an amplifier with excess gain at high frequencies -- no free lunch, you'll only increase the noise level up there. But probably also make an oscillator, because inverted impedances are tricky to use.
Alternately, you can tune out the capacitance to a certain extent (limited by the higher frequency details of the winding), but this only works for narrow bandwidth; for example, a 100-10kHz pickup might be tunable to 5k-15kHz or 10k-20kHz. Pretty useless for audio purposes.
Although given what you played with in your other thread,
https://www.eevblog.com/forum/projects/overkill-pikcup-combination/ maybe a multiband pickup wouldn't be too ludicrously out-there after all.

You'd tune it by simply putting an inductor in parallel, and adjusting the termination resistance as needed -- it'll want to be lower, like if the 10kHz case above is terminated with 100kΩ, tuned to 10k-20k it might need 50kΩ.
The response will simply look like a pickup EQ'd to a bandpass of whatever range it has. Apply appropriate EQ and mixing, and you've got the original signal back. Give or take differences in phase shift at the band edges, which can cause notches at the crossover points; a full stack could use phase shifters (maybe a 90 degree phaser pedal with fixed adjustment, rather than sweeping/phasing wah-wah), on each source, tweaked until the notches go away.
Hmm, shouldn't be too hard to compare frequency responses and make sure your pickups are flat (or within whatever response is desirable), you've got a sweep generator right there, just perform a slide.

The reductio ad absurdum could use all tuned pickups, with extra C to reduce the HF band limit, e.g. so you get a 60-300Hz band, then a 300-1kHz band, and... and mix them all together. Mind, it wouldn't do anything special -- it would just be amusing as an electromagnetic model of the human ear, of sorts. You could further subdivide the pickup channels into, say, single notes at a time -- and further, detect the amplitude of each channel, rather than its signal directly. Now you get a vocoder response! (But hey... perfect tuning every time?

)
Tim