The Haagerup inequalities are fundamental tools in the study of reduced group C*-algebras and allow one to compare the operator norm of convolution operators with much simpler L2-norms. The Haagerup inequalities were studied for free orthogonal quantum groups through the property of rapid decay (RD) within two different frameworks, namely `quantum RD’ and ‘twisted quantum RD’. The twisted quantum RD turned to hold for all (non-Kac) amenable orthogonal free quantum groups by Bhowmick, Voigt, and Zacharias in 2015. On the other hand, jointly with Brannan and Vergnioux, we proved that the twisted quantum RD does not hold for any non-Kac non-amenable orthogonal free quantum groups, whereas a weakened RD property is always satisfied. This weakened twisted quantum RD was improved recently and allows us to get suitable Haagerup inequalities with applications to obtain the optimal time for ultracontractivity of the heat semigroup.