The mineral linarite, PbCuSO4(OH)2, is a spin 1/2 chain with frustrating nearest-neighbor ferromagnetic and next-nearest neighbor antiferromagnetic exchange interactions. Our inelastic neutron scattering experiments performed above the saturation field establish that the ratio between these exchanges is such that linarite is extremely close to the quantum critical point between spin-multipolar phases and the ferromagnetic state. We show that the predicted quantum multipolar phases are fragile and actually suppressed by a tiny orthorhombic exchange anisotropy and weak interchain interactions in favor of a dipolar fan phase. Including this anisotropy in classical simulations of a nearly critical model explains the field-dependent phase sequence of the phase diagram of linarite,
its strong dependence of the magnetic field direction, and the measured variations of the wave vector as well as the staggered and the uniform magnetizations in an applied field.