I think the local peaks barrier is an artificial one due to us relying on ‘random’ landscapes, with this randomness resulting in computationally simple optimization problems. This seems like an unjustified assumption given that we don’t have any large (say on more than 20 genes) complete fitness landscapes, never-mind having an idea of what distributions over a family of fitness landscapes would look like. If we move away from randomly generating fitness landscapes then it is relatively easy to build families of hard landscapes where local peaks are hard to find: i.e. regardless of the (polynomial time computable) process we use to implement evolutionary dynamics, we can’t find even a local peak in polynomial. If we restrict our dynamics to just adaptive evolution or even further to fittest mutant or random fitter mutant then even rather simple seeming single-peaked landscapes can be difficult to find peaks in. See my preprint. for more info.
Unfortunately, in that example — similar to the LTEE — I am looking at fitness as the measure. It could be argued that the organisms aren't really getting more "complex" (they are just bit strings, after all) as they increase in fitness. But I still think we can learn a lesson from this for richer models: pick environmental challenges or fitness landscapes that aren't random but are instances of problems that are known to be hard for complexity classes like polynomial local search.

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