Excerpt from email:
Otmar Spinas 1/17/2019
I am writing as I came across your paper Hechler and Laver trees. I
noticed that your dichotomy for analytic sets there is implicit in the
characterization of strongly dominating analytic sets in the paper
"On tree ideals" by Goldstern, Repicky, Shelah and myself (Lemma 3.2).
If you look at property (4) in its proof, then if (4) fails the witness
F gives a Hechler tree whose branches are disjoint to set A.