A vanishing Yukawa coupling of the up quark could in principle solve the
strong CP problem. To render this solution consistent with current algebra
results, the up quark must receive an alternative mass contribution that
conserves CP symmetry. Such a contribution could be provided by QCD through
non-perturbative topological effects, including instantons. In this talk, we
present the first direct lattice computation of this topological mass
contribution, using gauge configurations generated by the Extended Twisted Mass
collaboration. We use the Iwasaki gauge action, Wilson twisted mass fermions at
maximal twist, and dynamical up, down, strange and charm quarks. Our result for
the topological mass contribution is an order of magnitude too small to account
for the phenomenologically required up-quark mass. This rules out the
"massless" up-quark solution to the strong CP problem, in accordance with
previous results relying on $\chi$PT fits to lattice data. The talk is based on
[Alexandrou et al., PRL 125, 232001 (2020)], where more details can be found.