The conjecture that the induced smash operation equals the Whitehead product

Let $X$ be a pointed space, let $n\geq 1$, and let $\widetilde{\theta}_{\iota}$ be the operation on homotopy groups induced by the distinguished class $\iota$. For $i,j\geq 1$, identify $\pi_{n+i}X$ with $\pi_i\Omega^nX$ and $\pi_{2n+i+j-1}X$ with $\pi_{n-1+i+j}\Omega^nX$. Whitehead-product conjecture. The operation $\widetilde{\theta}_{\iota}$ coincides with the Whitehead product $[-,-]$; equivalently, the diagram

\xymatrix{\pi_{n+i}X\times\pi_{n+j}X\ar@{=}[d]\ar[r]^-{[-,-]}&\pi_{2n+i+j-1}X\ar@{=}[d]\pi_i\Omega^nX\times\pi_j\Omega^nX\ar[r]^-{\widetilde{\theta}_{\iota}}&\pi_{n-1+i+j}\Omega^nX}

is commutative. A preceding proposition identifies the two operations after applying the Hurewicz homomorphism up to a possible kernel term, while the source leaves equality on homotopy groups as a conjecture.