I will argue that the experiment presented in the paper [1,2] actually supports Quantum Mechanics. This may be not quite explicit in the paper, but also there is nothing against the standard view on quantum mechanics in it.
Heisenberg originally stated his principle in terms of measurement-disturbance relationship (MDR). This is how he understood it at that time. The uncertainty principle which was proven theoretically, either in the context of wave mechanics, or from the non-commutativity of the operators, is correct, and it's correctness is acknowledged by the paper. This is called Heisenberg's uncertainty principle (HUP), and is very different from MDR.
The paper refers to previous theoretical works which disprove MDR, and present experimental evidence purported to confirm the violation of the MDR.
Why do I claim that the violation of MDR supports Quantum Mechanics? Because, if MDR would be correct, it would be enough to explain quantum uncertainty. Recall that even Heisenberg originally thought that the uncertainty is due to disturbance caused by measurement. If the states would behave as they are due to the measurement disturbance, then we could consider them classical, and extract Bohr's probability rule as we calculate probabilities in statistical mechanics. But we know this is not true. Quantum states exhibit properties which can't be explained by classical mechanisms. Among these, HUP plays an important role, together with entanglement. The service made by this paper is that it shows that the wrong version of the uncertainty principle can be violated. The authors seem to me to support the HUP:
These two readings of the uncertainty principle are typically taught
side-by-side, although only the modern one [HUP] is given rigorous proof.
Our work conclusively shows that, although correct for uncertainties
in states [HUP], the form of Heisenberg's precision limit is incorrect if
naively applied to measurement [MDR].
Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak Measurements
arXiv link to the paper
P.S. My answer is in contradiction with the interpretation given in the BBC article Heisenberg uncertainty principle stressed in new test. The BBC article is misleading, because it makes confusion between MDR and HUP.