Are direct search experiments sensitive to all spin-independent WIMP candidates?

The common analysis of direct searches for spin-independent Weakly Interacting Massive Particles (WIMPs) assumes that a spin-independent WIMP couples with the same strength with both nucleons, $\mathit{\text{i.e.}}$ that the spin-independent interaction is also fully isospin-independent. Though in a fully isospin-dependent interaction scenario the spin-independent WIMP-nucleus cross section is strongly quenched, the leading experiments are still sensitive enough to set limits 1-2 orders of magnitude less stringent than those traditionally presented. In the isospin-dependent scenario the difference between the limits of CDMS-II and ZEPLIN-I is significantly reduced. Here, a model-independent framework is discussed and applied to obtain the current general model-independent limits.