TL; DR - Two statistical/computer analyses take both sides, so we still don't know. However, given the advances in computing between 2009 and now, and the methods used, I would personally lean towards favoring the second paper that feels The Dark Tower was written by Lewis.
There are really only two actual academic instances that I can find discussing the attribution. (I am not counting the Lindskoog/Hooper mess, as that is squabbling, not scientific).
The first (arguing against it being a CS Lewis work), was published in the Austrian Journal of Statistics in 2009. They used a word analysis to compare Dark Tower (TDT) samples against Out of the Silent Planet and Perelandra. (These were chosen for thematic consistency). This paper concludes that TDT was not written by Lewis, or that he had a really bad day when doing so.
Ultimately, we saw moderately strong evidence differentiating the vocabulary in The Dark Tower from that of Out of The Silent Planet and Perelandra. While the “rare words” test again showed little discriminatory ability, the “slope” test and, to a greater extent, the “new words” test, rejected the hypothesis of Lewis authorship consistently more than would be expected.
Overall then, it does appear that word usage in The Dark Tower is inconsistent with that found in Out of The Silent Planet and Perelandra. This may be attributable to the fact that a manuscript in rough draft form (such as The
Dark Tower) would inherently display word usage different from a final, polished work or perhaps, as Poe (2007, p. 45) suggests, that The Dark Tower was written when Lewis had “. . . a bad day . . . [or simply] . . . committed a flawed plotline to paper”. However, it is also consistent with the claim that C. S. Lewis did not, in fact, write The Dark Tower.
Another computer analysis recently published (In the last couple of weeks) differs. Apologies, as I was unable to obtain the full article, but it compared TDT with works not only by Lewis, but Hooper and a known hoax letter by a third author. This comparison suggested that TDT was indeed written by Lewis.
Initial experiments found that the first six chapters of The Dark Tower were stylometrically consistent with Lewis’s known works, but the incomplete Chapter 7 was not. This may have been due to an abrupt change in genre, from narrative to pseudoscientific style. Using principal components analysis, it was found that the first and subsequent components were able to separate genre and individual style, and thus a plot of the second against the third principal components enabled the effects of genre to be filtered out. This showed that Chapter 7 was also consistent with the other samples of C. S. Lewis’s writing.
ETA: Lindskoog was an author who was convinced that Reverend Hooper (Briefly personal secretary to Lewis and later curator of his legacy in various capacities) forged many works, including TDT. She has been largely discredited, but people still believe her allegations:
This is a nice article exploring the allegations/divisions.