A query for my computational colleagues

There appears to be a consensus that natural languages are mildly context sensitive (MCS). Indeed, as I understand matters, this is taken to be an important fact about them and one that deserves some explanation. Here's my question: dos this mean that there is consensus that Kobele's thesis is incorrect? As I recall, it argued that NLs do not display constant growth given that the presence of Copy operations. Again, my recollection is that this is discussed extensively in Greg's last chapter of the thesis.  I was also led to believe that MCS languages must display constant growth, something incompatible with the sorts of copy mechanisms Greg identified (I cannot remember the language and I don't have the thesis to hand, but I am sure that you all know it). So, was Greg wrong? If so, how? And please use little words for this bear of little brain would love to know the state of play. Thx.