"Solved" isn't a term properly used in the sciences, and your quite legitimate confusion here is a nice example of why.
Science is the discipline of publicly testing ideas by systematic observation, controlled experiment, and Bayesian inference. It does not produce certainty, but rather knowledge. Unfortunately, because science is still a very young discipline (only three hundred years old) we have yet to really update our language to accommodate it, so we still talk in terms of "solution" and "proof" and the like, as if we were philosophers seeking after some chimeral goal like "certainty" or the ability to turn base metals into gold.
The questions scientists are interested in here are:
1) "Which is more plausible given the evidence we have: that we are computing something wrong in our Big Bang nucleo-synthesis calculations using existing physics; that our measurements of lithium abundance are wrong; that there is new physics that affects lithium production in the Big Bang; that our chemical evolution calculations are wrong for some reason; or that something else entirely is going on that we are missing?"
2) "What new evidence might we gather to clarify the situation given we currently don't have a stand-out idea that is sufficiently more plausible than the rest that no one can be bothered to do further investigations?"
Science is a human discipline, and as such is never "settled" except insofar as no on can be arsed to look at some question more deeply because the plausibility of the currently-best answer is so high (for example, while I think it very likely the Earth is heating up, I support further research like better satellite measurements of albedo: http://www.washington.edu/alum...)
With regard to lithium, we have a pretty good handle on Big Bang production assuming there is no new physics, but lithium has a number of characteristics that make it more strongly subject to the forces of what cosmologists call "chemical evolution"--the way the chemical composition of the universe changes through time due to stellar and other processes. The Korn et al work points to one particular way primoridal lithium could be hidden away. In the '90's there was similar work being done to show that various other processes could actually break lithium nuclei up over the course of the history of the universe.
Then there is also the problem that the whole "missing lithium" thing could be a result of a local anomaly in lithium abundance: after all, we have only sampled a small part of the universe. The work this /. post is about focuses on extending the reach of measurements to other galaxies, which is a start, although one could also imagine large-scale enrichment processes in the early universe that put us in a lithium-poor bubble, so no-doubt "additional work is required" to reach a sufficiently strong consensus that the missing lithium has been explained well enough to be not worth bothering with any more.