  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.
