The argument that there is a finetuning problem usually goes like this. The standard model implies that the observed Higgs mass is a bare mass plus quantum corrections. To make this concrete, we assume that the standard model is valid up to some energy scale Lambda; then we can talk about the bare mass and the quantum corrections as evaluated at Lambda. It turns out that the quantum corrections are enormous, but the observed mass is small, so the bare mass must be another enormous quantity finetuned so that the quantum corrections leave only the small observed mass.
This line of reasoning says nothing about what the physics above Lambda is. But S & W 2009 does present such a hypothesis: physics above Lambda consists of the standard model (or rather its nuMSM extension) coupled to asymptotically safe gravity. What I would like to do, is to spell out the argument that there is a finetuning problem, given this particular assumption about the UV physics.
Here's what I envisage. I would want to consider nuMSM, or SM or something simpler (like a toy model in which the S & W mechanism still works), coupled to the simplest truncation or concrete formulation of gravity for which asymptotic safety has actually been proven. That might allow us to be concrete about the graviton interactions that give rise to the quantum corrections in the standard model effective theory at scale Lambda.
An example of what I am talking about would be graviton exchange between two electrons. In the effective theory that should correspond to a new point interaction between two electrons, in the same way that Fermi's theory of the weak interaction can be obtained by integrating out the heavy weak gauge bosons of the standard model.
So if the true ultraviolet physics is nuMSM plus asymptotically safe gravity, the effective theory at scale Lambda shouldn't just be the standard model, it should be the standard model augmented with a large number of effective interactions due to graviton exchange, which will contribute to the quantum corrections to the Higgs mass.
I actually have no idea what happens if this analysis is carried through. Do you find that all of those effective interactions are inherently minuscule, because of powers of Lambda in the denominator, except for just those corrections at work in the scenario which predicts the Higgs mass? Or will we find that some type of finetuning magic will still be needed? Or is an analysis in terms of gravitons mistaken, and one needs to use some nonperturbative method?