Preconditioners are popularly used for speeding up Krylov subspace iterative method for solving linear systems from discretization of fractional differential equations (FDEs) on rectangular domains. Though some recent works have been developed for FDEs in general convex domains, it still has room for improvement in the design of preconditioner. For this sake, in this paper, we theoretically study the preconditioner problem for two-dimensional conservative space-fractional diffusion equations with variable coefficients and propose a robust preconditioner with penalty term which can deal with any convex domains significantly. We further prove that the proposed preconditioner equals to the coefficient matrix plus a low rank matrix and a matrix with small norm under certain conditions. It implies that the new preconditioner can effectively accelerate the convergence rate of Krylov subspace method. Experimental results show the good performance of the robust preconditioner.