In this paper from 2006, it states
In the present study, the existence of negative and positive feedback
regulation of ethylene biosynthesis is demonstrated in banana pulp and
peel tissues, respectively.
So, ethylene synthesis occurs in both the fruit and peel, and if there is any in the stem, it's likely irrelevant since the majority of the metabolically active tissue lies elsewhere.
In the comments to this answer, it has been questioned as to whether the lack of mention of ethylene synthesis in the stem means that it allows this possibility. As explained in the replies, ethylene is lighter than air, so if there is any production there, it can not affect the fruit of the banana as it will rise upwards. But there's another reference here to how ethylene synthesis occurs  which negates the possibility of ethylene synthesis in the stem:
Based on our results, ethylene production in ripening banana fruit can
be characterized as follows: (a) increase in the abundance of MA-ACS1
mRNA in the flesh is the first step of ethylene induction at the onset
of the climacteric; (b) the increased content of ACC in the flesh
induces a rapid increase of climacteric ethylene whereby ACC oxidase
is activated with the enhanced accumulation of MA-ACO1 mRNA once
And there's an Youtube video that busts this myth
And another bust
 Inaba A, Liu X, Yokotani N, Yamane M, Lu WJ, Nakano R, Kubo Y. Differential feedback regulation of ethylene biosynthesis in pulp and peel tissues of banana fruit. J. Exp. Bot. 2006 Dec 21;58(5):1047-57. doi: 10.1093/jxb/erl265. PubMed PMID: 17185740.
 Liu X, Shiomi S, Nakatsuka A, [..], Inaba A. Characterization of ethylene biosynthesis associated with ripening in banana fruit. Plant Physiol. 1999 Dec;121(4):1257-66. PubMed PMID: 10594112.