Not true. Genovese never asked for approval to hit Costello. After the fact, he claimed (falsely) that Gigante had shot Costello on his own, without Genovese´s consent. But how exactly he managed to save Gigante is not known. Anastasia however wanted war. Although no evidence of it, I believe the Commission met and solved the issue and thereby avoided a war, by approving the hit on Anastasia. Probably because the Commission members found Anastasia harder to restrain. At the same time, Genovese had support from Lucchese who had for a long time competed for influence in the same political sphere as Costello. So Lucchese was glad to see Costello gone. Anastasia, fatally, had no support on the Commission.