I've already asked this question on MSE (math.stackexchange.com/questions/885733/…) deeming it a not so hard one, but I haven't received enough hints to solve it. Please answer it there if you find it more suitable for MSE.

@AlexDegtyarev I couldn't track down your bound of 48 lines in neither of the articles mentioned. Segre treats this case at the end of his article, giving a bound of 64. Rams and Schütt in arXiv:1212.3511 give a similar proof in Lemma 5.5, producing a bound of 62.

I guess it is worth mentioning this article, again by Rams and Schütt, where the proof on the bound of 64 lines can be found arXiv:1212.3511. This is actually the article that inspired my question.

Ok, I found the article. It's "The density of rational points on non-singular hypersurfaces", II, Proc. London Math. Soc. 93 (2006), 273-303.