Kono's conjecture on Samelson products in p-regular exceptional Lie groups

Let $G$) be a $p$-regular exceptional Lie group, meaning that $G$ has the $p$-local homotopy type of a product of spheres. Write its type as $n_1,\ldots,n_\ell$, and let $\mathtt{t}(G)$ denote the set of type entries; for each $i\in\mathtt{t}(G)$, let $\epsilon_{2i-1}$ be the inclusion of the corresponding sphere into $G_{(p)}$. The notation $\langle\epsilon_{2i-1},\epsilon_{2j-1}\rangle$ denotes their Samelson product.

Kono's conjecture. For $i,j\in\mathtt{t}(G)$, there exists $k\in\mathtt{t}(G)$ satisfying

$$i+j=k+p-1$$

if and only if

$$\langle\epsilon_{2i-1},\epsilon_{2j-1}\rangle$$

is nontrivial.

The conjecture proposes a complete criterion for the nontriviality of Samelson products in $p$-regular exceptional Lie groups. The surrounding discussion records the known nontriviality result when $i+j=p+1$, while indicating that the general case was not determined there.