Sometime when I wasn't looking, the Wikipedia article on Satsugū (or "Kagoshima") dialect of Japanese got really, really good. Although maybe someone can help me out with this:

Additionally, the mid front unrounded vowel /e/ differs from standard Japanese in that it retains the Late Middle Japanese variation between palatalized [ʲe̞] and unpalatalized [e̞].

As I understand it, by LMJ (assuming we're talking about the central, dominant dialect), there may have been allophonic variation between palatalized and unpalatalized versions of /e/, but there was no phonemic distinction. In the Kagoshima dialect, on the other hand, there is (or was) a phonemic distinction, if a murky one. Because I am an aimlessly wandering autodidact, my original introduction to this was Komabashiri Shoji's "The Vowel e in Satsugu Dialect in the Eighteenth Century: Based on Ronichi tangoshu":

Ronichi tangoshu is a Russian-Japanese vocabulary, which is said to be written by Gonza, a man from Satsuma who drifted to Russia by boat in the 18th century. In the vocabulary, two letters were used to represent the vowels which correspond to /e/ in the present-day Satsugu Dialect : 'e' and 'ѣ', for example, <eдa> (eda 'branch'), <ѣкаки> (ekaki 'painter'). In this paper, by examining the use of these two letters in Ronichi tangoshu, the Russian orthographical system at that time and the phonological system in the present-day Satsugu Dialect, I conclude that there were phonological distinctions between /e¹/ and /e²/ in the syllables e (at word-initial position), ke, ge, se, ze, te, de, ne, be and me in the Satsugu Dialect in the 18th century, and one or both of /e¹/ and /e²/ were palatalized. This conclusion is significant for the reconstruction of the history of the Japanese phonological system.

As the paper explains, a lot of the variation can be explained diachronically within Kagoshima dialect (resulting from vowel coalescence and so on), but there are too many exceptions to sweep under the rug.

To summarize, as far as I can tell, (a) whatever variation existed between palatalized and unpalatalized /e/ in LMJ was different from the behavior of /e/ in Kagoshima dialect, and (b) given that, if Kagoshima dialect is retaining anything, it must be from an even earlier stage of Japanese. So I don't know quite what to make of "retains the Late Middle Japanese variation between palatalized [ʲe̞] and unpalatalized [e̞]".

However, I cheerfully admit to knowing less about Kagoshima dialect, and quite probably MJ phonology, than whoever did the work on this article, not to mention many of the people reading this. Can anyone set me straight?