Martin's Axiom and Weak Kurepa Hypothesis

I show that it is consistent relative to the consistency of a Mahlo cardinal that Martin’s axiom holds at ${\omega }_2$, but the weak Kurepa Hypothesis fails. This answers a question posed by Honzik, Lambie-Hanson and Stejskalová. The consistency result is obtained by constructing a model where the weak Kurepa Hypothesis fails in any c.c.c. forcing extension.