According to the wikipedia page, there are two preconditions necessary for lightning:
Firstly, a sufficiently high potential difference between two regions
of space must exist, and secondly, a high-resistance medium must
obstruct the free, unimpeded equalization of the opposite charges. The
atmosphere provides the electrical insulation, or barrier, that
prevents free equalization between charged regions of opposite
Since the "string of pearls" are massive counterclockwise rotating storms, one would presume that high potential differences would exist in these turbulent regions. So the only remaining reason for not having lightning would be if there was an electrically conductive atmosphere. But there are numerous examples of lightning strikes in storms in Jupiter, so, unless the atmosphere at the Pearls is somehow different than the rest of Jupiter, one would expect lightning there as well. Indeed, "storms on Jupiter are always associated with lightning".
From NASA's website, here is a diagram of one of the mechanisms thought to be responsible for lightning on Jupiter:
In conclusion, if you want a reason not to have lightning at the string of pearls, there would need to be little or no turbulence or an an electrically conductive atmosphere. Since neither of these conditions is the case, one could expect lightning at these storms.