The J-homomorphism conjecture on cohomology indecomposables

Let $J$ be the $J$-homomorphism, let $\beta_j$ denote the indicated cohomology classes associated with $SO$, and let $obreak\gamma_{1[1]},\gamma_{1[2]},\gamma_{2[1]}$ denote the cohomology indecomposable generators in the Nakaoka-module presentation used in the paper. J-homomorphism conjecture. The induced map on cohomology indecomposables satisfies

$$\gamma_{1[1]}^{2k+1}\mapsto\beta_{2k+1}$$ $$\gamma_{1[2]}^{i}\gamma_{2[1]}\mapsto\beta_{2i+3}$$

and all other indecomposables map to $0$. The displayed formulas are supported by calculations through degree $15$, but the general assertion is presented as a conjecture.