Is there a serious effort underway to move from a highways-in-the-sky approach to a mesh-network approach?
Currently SESAR (Single European Sky ATM Research) works on implementing a similar system in the European Union.

As stated on the website :


  SESAR’s vision builds on the notion of trajectory-based operations’ [...] meaning that aircraft can fly their preferred trajectories without being constrained by airspace configurations.


The aim is to improve capacity in the European sky and be more flexible among the actors too (like temporary closed airspace for military training, handle volcanic eruptions and disruptions of the airspace).

SESAR has different projects to address the needs of imporving Air Traffic Management in Europe that you can find on their website.

Edit

There are current concrete solutions from SESAR and EUROCONTROL (the European organisation for ATM in Europe) to make this happen at the moment in the European airspace, such as the Free Route Airspace with set target of implementation (that we could consider "serious effort") :


  By the end of 2017, 51 ACCs have either fully or partially implemented Free Route Airspace operations [...] By the end of 2019, most European airspace is expected to have implemented FRA, with all airspace having this type of operations by 2021/2022.