  We consider the Laplace operator in a tubular neighbourhood of a conical
surface of revolution, subject to an Aharonov-Bohm magnetic field supported on
the axis of symmetry and Dirichlet boundary conditions on the boundary of the
domain. We show that there exists a critical total magnetic flux depending on
the aperture of the conical surface for which the system undergoes an abrupt
spectral transition from infinitely many eigenvalues below the essential
spectrum to an empty discrete spectrum. For the critical flux we establish a
Hardy-type inequality. In the regime with infinite discrete spectrum we obtain
sharp spectral asymptotics with refined estimate of the remainder and
investigate the dependence of the eigenvalues on the aperture of the surface
and the flux of the magnetic field.
