We prove a super logarithmic lower bound on the Conon deterministic communication complexity of the Clique vs. Independent Set problem introduced by Yannakakis (STOC 1988, JCSS 1991). As a corollary, this implies super polynomial lower bounds for the Alon -- Saks -- Seymour conjecture in graph theory. Our approach is to first exhibit a query complexity separation for the decision tree analogue of the UP vs. coNP question -- namely, unambiguous DNF width vs. CNF width -- and then "lift" this separation over to communication complexity using a result from prior work.