alt Hacker NewsI wouldn't say that we have no experimental data which contradicts them. Rather, we do have experimental data which contradicts them, but no experimental data that points us in the direction of a solution (and whenever we go looking for the latter, we fail).
Consider e.g. neutrino masses. We have plenty of experimental data indicating that neutrinos oscillate and therefore have mass. This poses a problem for the standard model (because there are problems unless the mass comes from the Higgs mechanism, but in the standard model neutrinos can't participate in the Higgs mechanism due to always being left-handed). But whenever we do experiments to attempt to verify one of the ways of fixing this problem -- are there separate right-handed neutrinos we didn't know about, or maybe instead the right-handed neutrinos were just antineutrinos all along? -- we turn up nothing.
> the standard model neutrinos can't participate in the Higgs mechanism due to always being left-handed
This again? It's only true if you insist on sticking with the original form of Weinberg's "model of leptons" from 1967 [1], which was written when massless neutrinos were consistent with available experimental data. Adding quark-style (i.e. Dirac) neutrino mass terms to the Standard Model is a trivial exercise. If doing so offends some prejudice of yours that right-handed neutrino can not exist because they have no electric and weak charge (in which case you must really hate photons too, not to mention gravity) you can resort to a Majorana mass term [2] instead.
That question (are neutrinos Dirac or Majorana?) is not a "contradiction", it's an uncertainty caused by how difficult it is to experimentally rule out either option. It is most certainly not "a problem for the standard model".
[1] https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.19.1264
[2] https://en.wikipedia.org/wiki/Majorana_equation#Mass_term