They have performed two measurements on Alice's photons and claim that the first measurement is "an intervention on A [that] forces the variable to take a specific value".
I have seen plenty of posts here on PF saying that first measurement breaks entanglement and after it particle is not entangled any more with the other one. So this is the statement that is tested in this experiment as I see it.

The experiment rules out a large class of non-local causal theories that make different predictions than quantum mechanics. But it does not rule out all non-local causal theories. In particular, it certainly does not rule out Bohmian mechanics.

In short, Bohmian mechanics is a non-local causal theory constructed so that it makes the same measurable predictions as standard QM. Since the experiment above certainly does not violate predictions of standard QM, it follows that it also does not violate predictions of Bohmian mechanics.

Well, maybe I misunderstood it, but I was almost sure that Bohmian mechanics pretends to explain correlations such as the breaking of the Bell Inequality by means of causal influence from one measurement outcome to the other. However, at the level of measurements of single pairs such inequalities cannot be broken, and the experimenters here claim that their results "demonstrate that a causal influence from one measurement outcome to the other[..] cannot explain the observed correlations". And: "Our results highlight the incompatibility of quantum correlations not only with the well-known Bell-local causal models but also with nonlocal causal models, where one measurement outcome may have a direct causal influence on the other."
To me that looks like a strong disagreement. Indeed, as so often in this forum, I'm mystified - so I'd like to be "demystified".

Could you, by any chance, clarify in detail (minimal event by event analysis) how Bohmian mechanics pretends to pull off the trick of breaking Bell's inequality in practice? For example with the CHSH inequality as given by Sica, which, I think, corresponds to that of Bell in appendix 2 of his Bertlmann's socks paper:
|<ab> + <ab′>| + |<a′b> − <a′b′>| ≤ 2
- https://cds.cern.ch/record/142461/files/19

In the paper, the authors say:
"We focus on the class of models that satisfy causal parameter independence ..."
Therefore, their statements you quoted above should have additional disclaim "... provided that parameter dependence is absent" or something similar.

Regretfully that page does not show up at my (x,y,z,t)... However the preceding page does show up. After explaining the meaning of "parameter independence", the book states that "quantum theory [..] satisfies Parameter Independence"... Also, I regret that still no way is presented how Bohmian mechanics could break Bell's Inequality. But that's ok, we are all here in our free time and Bohmian mechanics is only a subtopic of this thread, so I won't insist. Hopefully there will be more interesting comments on the "no nonlocal causality" experiment.

Since parameter independence is the crucial assumption in the paper we are discussing here, let me demystify parameter independence (PI). If A and B are two spatially separated systems, PI means that nothing in A depends on experimental setups ("parameters") in B. (And vice versa, with A and B exchanged.) In other words, PI is the same as absence of non-local contextuality.

Now, standard QM obeys PI, while Bohmian QM does not. Yet, they are experimentally equivalent. How that can be? That's because "nothing in A" does not have the same meaning in standard QM and Bohmian QM. In the Bohmian case, "nothing" includes the hidden variables, which are absent in standard QM. So to reproduce predictions of QM in terms of causal hidden variables (which the discussed paper is about), the paper shows that causal hidden variables (CHV) cannot work if CHV obey PI. In other words, they have shown that local contextuality at the level of CHV is not consistent with QM and experiments. When put in this form, it does not look so much new and surprising.

Needless to say, Bohmian mechanics (as a nonlocal CHV theory) does not obey local contextuality at the level of CHV.

Let me ask a B level question: How does/can quantum theory satisfy parameter independence ,ie locality. According to Bell QM is parameter dependent:/quantum contextual.* " Statistical predictions of QM are incompatible with separable predetermination. Incompatible in sense in which parameters are added to QM to determine results of individual measurement. There must be a mechanism whereby the setting at one measuring device (a) can influence readings of another (b) however remote,more ever the signal involved must propagate instantaneously.
* Quantum contextual : The result of measurement of observable (a) depends on another measurement on observable (b)
So it seems that in order for contextual hidden variables to support QM predictions they have to be non local, parameter dependent . quantum contextual ?
Can there be non local affects/ non local causality that produce quantum correlations in above cases
that do not involve superluminal signals ?

Yes. All non-local correlations measured so far are such that they do not involve superluminal signals. (By a "signal", one means a transmitted message freely chosen by a human.)