Hello, paleobiologist here. The short of it is that this particular work is not strongly relevant to Punc-Eq, but other works Lenski has done are relevant.
Well, Lenski's work (particularly the work he did with Travisano) has been part of the Punc-Eq debate for the last ten years. Basically, Lenski, Travisano and a few others showed (around 1995, in both a PNAS and a Science paper) that isolated bacteria populations placed on different substrates (nutrients) would rapidly adapt to their new substrate and followed by little apparent change (i.e. phenotypic change, in this case cell size and shape).
Sounds like Punc-Eq, right? Rapid change followed by stasis.
Now, the important thing to understand about Punc-Eq is that other than a few particularly closed-minded population geneticists, really no one has disputed that the phenomenon of Punc-Eq occurs: that evolutionary change is heterogenous, particularly over long time spans. There are some important exceptions, but it is now well accepted that (for the most part) the fine-scale fossil record consists mostly of rapid shifts, stasis and random-walks (G. Hunt, 2006 in PNAS). Also, it has been well documented that populations under adaptive change in the wild, like Darwin's Finches, can show rapid change.
The real argument is about cause. Lenski's work (and the work of others, such as Russell Lande) argued that Punc-Eq could be explained by environmental shifts, resulting in rapid adaptation followed by stabilizing selection (resulting in stasis). Gould and Eldredge argued multiple causes for Punc-Eq, but Gould tended to favor developmental constraint (at least in the 1980's); that organisms had difficulty changing their body-plan except during speciation when something changed in their population dynamics. Essentially, the developmental path became canalized and organisms couldn't jump out of it without some special situation. Eldredge argues (more along Sewall Wright lines) that interbreeding populations strung out across different habitats will stop the species from adapting to any one particular habitat. It is difficult to test either the intrinsic constraint hypothesis or the geographically-dispersed populations one. Thus, the selection-based hypothesis is well supported by evidence because it has been tested a bunch of times and the other two just don't get tested.
The real message of this article by Barrick et al is about neutral evolution: Kimura's 1970s hypothesis that the evolution of the genome was mostly by chronologically regular neutral mutations (changes to the genome that didn't result in changes to fitness). This is the basis of the molecular clock hypothesis. It may be important, as I used to get this mixed up myself, to point out that mutation rate is just the rate at which all genetic changes are introduced; thus adaptations are beneficial mutations. Its important to Kimura's hypothesis that the clock be due to neutral (or slightly deleterious) mutations, because otherwise it means the mutation rate is being controlled by adaptation, which is controlled by that vague and rapidly changing environment (which is a big problem for the molecular clock).
It appears that once the rapid adaptive changes had occurred in Lenski's bacteria populations, adaptation rate decreased, mutation rate decreased and was very clock-like for the next 20,000 generations. Sounds like Kimura was right, eh? Problem is, it appears those mutations were actually beneficial mutations, not neutral mutations. So apparently that was a period of very slow adaptation... genetic fine-tuning to the environment, perhaps. After 20,000 generations, the mutation rate suddenly sky-rocketed and a whole bunch of neutral mutations appeared. The big picture? The mutation rate was so heterogeneous and the signal of adaptation so hard to piece out, that reconstructing genomic rates of evolution from extant organisms (using the molecular clock hypothesis) may be problematic without lots and lots of data. And that's a pretty big message.