I'll be updating this answer over the course of time, these are just things I could think of right now.

**String theory **

- Formulating M(atrix) theory and Matrix string theory on a general background -- Perturbative string theory can be formulated on a general metric as is taught in any introductory string theory book or course, but the non-perturbative formulation of M-theory, as well as its compactifications Heterotic
*E*8^{2} and Type IIA, are formulated only on flat spacetime. From what I know these also aren't "exact" cases of AdS/CFT, in that they aren't actually formulated on AdS.
- "Solving" the landscape -- a phenomenological problem, that is finding the right vacuum in the landscape. According to Denef and Douglas (2006) - hep-th/0602072, the problem is NP-hard. Here's a criticism of the paper by Lubos Motl.

(John Schwarz also has his own review paper on the issue here)

**Loop Quantum Gravity**

LQG is pretty messed-up to say the last, especially on the issue of Lorentz symmetry, because the spacetime is discrete, and the Lorentz transformations on the discrete units would then be distinguishable. The easiest way to test this is to test for a difference in speed between high- and low- frequency lights, and in precisely this way the level to which it breaks Lorentz invariance has already been falsified, leaving string theory as the only BSM theory unfettered. Unless LQG manages to go under a radical reform so that the spacetime is no longer discrete -- and many would argue that such a theory is no longer LQG, because the discreteness can be considered a fundamental postulate of the theory (it was derived, but it's basically equivalent to the defining of the Ashtekhar variables) -- you can consider it falsified.