This work addresses the fundamental question concerning the nature of quantum states, specifically to what extent they represent an agent’s knowledge about the system, that is, is epistemic in nature. We show that, despite the existence of the Kochen-Specker model, which is a maximally $\psi -$epistemic model for single-qubit systems when projective measurements are employed, no maximally $\psi -$epistemic model can reproduce statistics arising out of non-projective measurements on a pair of single-qubit states. This is demonstrated in an experimentally robust way using only two pure single-qubit states and their equal mixture, by constructing a two-input, three-output game called (RAD) and generalizing the notion of epistemic and quantum overlaps using RAD. Furthermore, we maximize the difference of generalized overlaps instead of minimizing their ratio, and in the process point out why that is a stricter criterion for refuting $\psi -$epistemic models.