How closely can string theory mimic asymptotic safety for gravity? Asymptotic safety in quantum gravity has various problems, but (as I see it) it has two points in its favor. 
First, the hypothesis has apparently passed a significant test which skeptics predicted it would fail (see the 2016 paper cited by Arnold Neumaier). 
Second, it was successfully employed to predict the mass of the Higgs boson. As Daniel de França ("MTd2") has remarked, that might be the first confirmed prediction of a quantum gravitational theory!
So I am wondering whether string theory can mimic asymptotic safety. For example, could there be a stringy regime in which gravity is described by one of the truncations of Einstein gravity that has been shown to be asymptotically safe? 
 A: 
So I am wondering whether string theory can mimic asymptotic safety.

it's worth noting that all developments in string theory converge to the observation that fundamentally gravity is not a local field theory, while the claim of asymptotic safity is just the contrary, that it is a local field theory at arbitrary high energy.
Within perturbative string theory itself, gravity is carried by the closed string sector, which fundamentally is not a local field theory, but a string field theory, non-local at about the string scale. 
Within non-perturbative string theory, gravity is argued to be holographically dual to a local and conformal field theory (AdS/CFT), and as such is not itself a local field theory.
But also apart from the a priori assumption that quanta of gravity a carried by strings, people see the scaling behaviour of Bekenstein-Hawking entropy and conclude that if this entropy is to be carried by quantum degrees of freedom, then this cannot be those of of a local field theory. The AdS/CFT correspondence is in a way a grand generalization of this phenomenon.
A quick, clean, informative  and concise account of this is here:


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*Assaf Shomer, "A pedagogical explanation for the non-renormalizability of gravity" (arXiv:0709.3555)


see in particular the last little section IV.
You might also enjoy googling for "Jacques Distler" "Asymptotic safety" to see what one particular string theorist has to say in detail about the AS claims.
A: There is now a paper exploring the possibility of an asymptotically safe regime just below the field-theory limit of string theory. As a basic test of plausibility, they check for compatibility with the weak gravity conjecture. Whether the higher-derivative gravitational terms descended from string theory are compatible with asymptotic safety, should also be a key issue to investigate. 
