Let $M$ be a surface. For any probability measure $\mu$ on the space of $C^2$ diffeomorphisms on $M$, and verifying certain conditions ("hyperbolicity" and "almost volume preserving"), we show that any $\mu$-stationary SRB measure is absolutely continuous with respect to the ambient Haar measure. The proof is motivated by the "transversality arguments" in Tsujii's work on partially hyperbolic endomorphisms. This talk is based on a joint work in progress with Aaron Brown, Davi Obata and Homin Lee, which is a stronger version of a previous work (https://arxiv.org/abs/2409.18252) by the same authors..