In an earlier paper, a new class of thin-shell cavities were proposed to evade the steep frequency scaling of conventional axion haloscopes. In this follow-up work, we see that a generalized conic geometry enables robust frequency-tuning for these large-volume cm-wave cavities. The frequency-defining dimension of a conic shell-cavity changes symmetrically and uniformly during tuning, maintaining a high axion coupling efficiency (the form factor) to an external solenoid field. It is further shown that such tunable geometry is not restricted to circular cones. A general prescription for arbitrary volume-filling conic shell-cavities is developed and direct solutions are obtained for the created numerical models. The largest of the realized designs is a meandering brain cavity that is tunable over a frequency range of 20%. The scan rate of this cavity is three orders of magnitude larger than that of a scaled cylindrical cavity used in the current generation experiments. The prospects for such a large improvement in the scan rate should motivate R & D efforts in fabrication and other implementation techniques. If these engineering challenges can be met, cavity-based axion haloscopes can stay competitive at frequencies higher than a few GHz. We propose an experimental configuration at 20 GHz (~ 80 $mu$eV) using an array of brain cavities and compare it with other proposals for similar frequencies.
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