A II$_1$ factor $M$ is called existentially closed if any larger II$_1$ factor can be realized as an intermediate subfactor of the inclusion of $M$ into one of its ultrapowers. I will discuss a new result showing that if a II$_1$ factor $M$ is existentially closed, then every $M$-bimodule is weakly contained in the trivial $M$-bimodule or, equivalently, every normal completely positive map on $M$ is a pointwise limit of maps of the form $x\mapsto\sum_{i=1}^ka_i^*xa_i$, for some $k\in\mathbb N$ and $(a_i)_{i=1}^k\subset M$. I will also discuss an operator algebraic presentation of the proof of the existence of existentially closed II$_1$ factors. While existentially closed II$_1$ factors have property Gamma, by adapting this proof we are able to provide examples of non-Gamma II$_1$ factors which are existentially closed in a large class of extensions. This talk is based on joint work with Hui Tan.