How can phase separation be used for kinetic proofreading?

T cells are important immune system cells which function by identifying proteins present on other cells, mediated by their T cell receptors (TCRs). These receptors are incredibly specific, having low false positive and false negative rates. Recent work suggests that an energetically driven phase separation process between phosphorylated and unphosphorylated versions of the membrane-bound protein LAT may play an important role in this. This motivates the theoretical question: under what circumstances could such a phase separation process give rise to a proofreading mechanism that can achieve the kind of specificity observed in TCRs? We propose a theoretical model which captures important characteristics of pLAT condensation, and determine what ingredients are required to allow a phase separation mechanism to have good kinetic proofreading performance. In particular, we focus on receiving operating characteristic curves to measure the performance of different mechanisms, allowing us to examine each mechanism at the timescale of interest to the T cell. Beyond their applications to the TCR, our analysis reveals general principles about the implementation and performance of kinetic proofreading via phase separation in biochemical systems.