In 2013, I and Yanir Rubinstein introduced special class of log Fano varieties which are known now as asymptotically log Fano varieties. We classified all of them in dimension 2 (asymptotically log del Pezzo surfaces) and studied the existence of Kahler-Einstein edge metrics on them (K-stability).
Recently we almost completed this study together with Kewei Zhang. In my talk, I will explain what we did and why our original expectation is not correct.