The appropriate amount of sympathy (and I'm pretty sure you know this) is to not say you don't give a shit about them and is simply to give a shot about them. You don't have to tweet or mourn them but actually care a little bit about the unfortunate circumstances they are in. Shut up about virtue signalling, taking about that like I'm on the left and want you to do it, I'm not gonna argue with someone who instantly becomes facetious. False sympathy? If  what I just described is false sympathy to you then I was correct with my original statement. You are unempathetic.