user76284 @: The analytic continuation of a given function is (more or less) unique, but in this context we not not HAVE a function. We have to chose it and the number of choices is pretty much unlimited. For a reference to the above, see e.g. Hardy, Divergent series, Theorem 9 (p. 52 in the edition I am looking at).

@skupers. Can you elaborate on the conclusion? We need (1) An analogous invariant in $\pi_3(STOP(4))$ (even though it would be trivial in the end) and (2) a commutative diagram. I am not sure about either.

Thank you. So this is where the mistake was. By the way, in the Remark I mentioned Mahowald have really said "it can be shown that $\theta_4^2=0$". And yes it can, except this was actually done 40+ years later! (Theorem 1.2 in this paper of Xu.) Amazing. I am still curious though why Shtan'ko counted the problem as solved in 1970s. Was there a time when experts were optimistic about it?

In case anyone is curious, they exist link.springer.com/article/10.1007/s00039-019-00476-6

I think this description is good enough for me. (And now I do not mind moderators closing my question as a duplicate.)

Thank you. What I want is to have a picture of the geometry. (Even if only a bit. Then, at least, I would not have to completely trust a topologist that all of this works.) To do this your way I would need to make sense of (1) the map $S^8\to S^5$ in $\pi_8 S^5$ and of (2) the geometry of $\pi_8 S^5\to \pi_9S^3$. Both things look rather obscure to me (even if I more or less understand the argument on the formal level).