Cowling-Haagerup constant of the product of discrete quantum groups

We show that (central) Cowling-Haagerup constant of discrete quantum groups is multiplicative, which extends the result of Freslon to general (not necesarilly unimodular) discrete quantum groups. The crucial feature of our approach is considering algebras $\mathrm{C}(\mathbb{G}),{\mathrm{L}}^{\mathrm{\infty }}(\mathbb{G})$ as operator modules over ${\mathrm{L}}^1(\mathbb{G})$.