The Whitehead conjecture for weak S-homotopy equivalences of globular CW-complexes

Let $f$ be a morphism of globular CW-complexes from $X$ to $Y$. A weak S-homotopy equivalence is a morphism that induces a bijection between the $0$-skeleta and, for every $\alpha,\beta\in X^0$, a weak homotopy equivalence

$$\mathbb{P}(X,\alpha,\beta)\longrightarrow\mathbb{P}(Y,f(\alpha),f(\beta)).$$

An S-homotopy equivalence is a morphism having an inverse up to dihomotopy. Whitehead conjecture. The following assumptions are equivalent:

1. $f$ is a weak S-homotopy equivalence.
2. $f$ is an S-homotopy equivalence.

This is the converse of the result that every S-homotopy equivalence is a weak S-homotopy equivalence. The source notes that the conjecture was later solved, so the equivalence is no longer open.