Shaposhnikov and Wetterich assumption is no new physics, no GUT physics and AS, they get 126 to within 1 gev uncertainty, and the latest results of proton decay seem to support this.

also, as the LHC thus far found no evidence of SUSY, higgs compositeness, extra dimensions, or conformal there's this paper

Gauge hierarchy problem in asymptotically safe gravity--the resurgence mechanism Christof Wetterich, Masatoshi Yamada
(Submitted on 9 Dec 2016)
The gauge hierarchy problem could find a solution within the scenario of asymptotic safety for quantum gravity. We discuss a "resurgence mechanism" where the running dimensionless coupling responsible for the Higgs scalar mass first decreases in the ultraviolet regime and subsequently increases in the infrared regime. A gravity induced large anomalous dimension plays a crucial role for the required "self-tuned criticality" in the ultraviolet regime beyond the Planck scale.
Comments: 5 pages, 1 figure
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1612.03069 [hep-th]

Shaposhnikov and Wetterich predicts 126 gev higgs boson mass, no new physics from fermi scale to planck scale and also predicts the higgs is stable under asymptotic safety for quantum gravity, with no need for susy, technicolor, conformal or extra dimensions

can the other proposals that predict 126 gev higgs boson mass also explain other observations and challenges?

Please don't misrepresent the predictions.
In 2009, they predicted 126.3 ± 2.2 GeV, based on "Assume that gravity is asymptotically safe, that there are no intermediate energy scales between the Fermi and Planck scales, that the gravity induced anomalous dimension of the Higgs selfcoupling is positive"
Also in 2009, they predicted 150 ± 24 GeV, based on "Assume that gravity is asymptotically safe, that there are no intermediate energy scales between the Fermi and Planck scales."

Both are "no new physics", and the second prediction is compatible with the whole mass range allowed by electroweak precision fits.

Probably not all, but certainly some of them. Taking the theory uncertainties as standard deviation (they are not), there are 24 predictions within 1 sigma, and 33 within 2 sigma of the current world average of 125.09 (+- 0.2x) GeV. I didn't include predictions with an uncertainty of 50 GeV or more. Do you really think all apart from this one got excluded?

The hype around the Shaposhnikov and Wetterich paper started in July 2012, at that time nothing apart from the Higgs mass got added. But it is easy to point to a specific paper and say "see, there was a reasonable mass estimate", while ignoring (a) tens of other papers with wrong but equally justified mass estimates and (b) tens of other papers with compatible mass estimates.

They were merely pointing out that the predicted Higgs mass depends on the sign of the gravity induced anomalous dimension. If its positive, one gets the 126 prediction; and if its negative, one gets the less certain 150 prediction. But as they made clear in the abstract: "The case ##A_{\lambda }> 0## is favored by explicit computations existing in the literature." https://arxiv.org/abs/0912.0208

i was going to mention that the result is consistent with "gravity induced anomalous dimension of the Higgs selfcoupling is positive"

the result then is evidence of the assumption of

"that there are no intermediate energy scales between the Fermi and Planck scales"

which thus far to date, lhc, electron edm, proton decay and dark matter searches currently favor

"that the gravity induced anomalous dimension of the Higgs selfcoupling is positive"

is positive

Wetterich also argues that under AS, the higgs by itself is stable, with no additional physics such as susy or technicolor needed. thus far lhc results seems to support this claim, and it's fairly minimal

some of the others papers are based on SUSY which have not yet been observed, or other physics which have not been observed. and 126.3 ± 2.2 GeV seems to be the tightest fit.

i.e a paper that is based on low energy susy and predicts 140 gev ± 20 gev is "right" but the susy is not supported and its margin of error is much higher than S&W paper.

https://arxiv.org/abs/1701.02311 Hypercuboidal renormalization in spin foam quantum gravity Benjamin Bahr, Sebastian Steinhaus
(Submitted on 9 Jan 2017)
In this article we apply background-independent renormalization group methods to spin foam quantum gravity. It is aimed at extending and elucidating the analysis of a companion letter, in which the existence of a fixed point in the truncated RG flow for the model was reported. Here we repeat the analysis with various modifications, and find that both qualitative and quantitative features of the fixed point are robust in this setting. We also go into details about the various approximation schemes employed in the analysis.

one potential drawback is breaking GR diffeomorphism invariance