I'm not following, paleo-observations of solar magnetic fields?Or there may be little causal connection of any kind.
No, the mechanism was how it could affect clouding. The clouding would then have to affect the temperature. Are solar magnetic fields the dominant factor that controls clouding on Earth, and is clouding the dominant effect in the temperature? No, on both counts. So the hypothesis, apparently, is that even though there are far more important factors that control clouding and temperature over "paleoclimatic" timescales, somehow the tiny influence of solar magnetic fields is the crucial variation we must focus all our attention on? A conclusion of so unlikely a nature requires very solid evidence, and must go way beyond "this is how one might possibly imagine such a connection". The physical mechanism cannot just be a cartoon-- it must be established by demonstrable causality, not pure correlation and speculation. The speculation by itself is neither right or wrong, it is just a speculation. It is the followup that is lacking.
These predictions were not post-facto? The "theory" was not set up to explain existing data? Because you must realize that weather pattern prediction is a rather important industry, and if a crucial term (like cosmic rays) are being left out, the weatherman needs to know.
Of course it has not been resolved, the observations that could lend some solid support haven't been done! But the problem is, the burden of proof lies on the person claiming the correlation is a causation, it is never sufficient to say "we'll believe it until observations prove us wrong". Not when important policy decisions are at stake, that's just an apologetic for ignoring the evidence that other observations have already accrued. Granted, I commend these researchers for not just saying "I'm not convinced about the other arguments", they are actually trying to put new testable arguments on the table. But putting them on the table by noticing some correlations is just the very first step, and cooking up obscure and untested mechanisms to support it is the second-- what is then required is the demonstration that the mechanism is at play with direct observations of that very mechanism at work. Not indirection correlations, direct measurements of the causal agent and the proposed response. When that important final step is lacking, it's all just pure speculation based on correlation.
Not just that, the planet should show increased clouding due to whatever is the putative transfer agent to the Earth! This is my point, if the mechanism says "magnetic field A causes particle B to stream to Earth and generate cloud C which then causes cooling D", a scientific test cannot simply show A and D, it must also show B and C-- especially when B and C are perfectly easy to observe independently of A and D. That's what I am not seeing here-- a direct scientific test, not just a plausibility argument based on a post-facto correlation.
No, if the hypothesis is not correct, you can say the change will be coincidentally related, you cannot say there will be no change. One always has to assess the coincidence probabilities, especially when has a range of possible correlations to look for. That's why the direct tests are what you need.
Those predictions are not based on models, as far as I can see, they are based purely on empirical extrapolations of past results. And good researchers should always remember the ubiquitous stock-market warning: "past performance is not necessarily predictive of future results". I wouldn't bank any serious money on any of those cycle predictions, to be quite frank-- expressly because there is no known physical model for determining solar magnetic activity. Even the most basic aspects of the solar cycle are not understood concretely.
And this so-called "adjustment" is based on what independently demonstrated physics? I smell a big rat in that word. You cannot demonstrate that electroscavenging occurs in the same observation that you correct for its effects on the thing you are actually observing, you need a direct observational test on the proposed mechanism or it is underconstrained. And anyone can do pretty much anything with underconstrained data.
Earthshine on the Moon is a very imperfect way to test the cloud cover of the Earth, because the Moon only samples a single direction, so any variation in the reflected brightness with angle is pure noise in that experiment. Also, one needs to use consistently controlled observations of that Earthshine, which might be quite hard to do at the required accuracy level (I can't comment on the error analysis, I'm just saying that needs to be looked at very carefully, and I wager has been, by the paper's detractors). The red flag I always look for in post-facto correlations is that the effect is more apparent in noisy data than it is in cleaner data.