Conjectural formula for the derivative of the homomorphism sigma

Let $p':= p-b$ and $q':= q-a$. For $\xi_1\in\pi_pSO_{q'}$ and $\xi_2\in\pi_qSO_{p'}$, let $h\colon\pi_iSO_j\to\pi_i(S^{j-1})$ be induced by $SO_j\to S^{j-1}$, let $\partial\colon\pi_{m+1}(S^k)\to\pi_mSO_k$ be the boundary map, and let $\overline\xi_{1*}$ and $\overline\xi_{2*}$ be the homomorphisms defined using the suspended $J$-homomorphisms and the inclusions into $SO_{p'+q'-1}$. Conjectural formula. Up to sign, the homomorphism

$$d_*\circ\sigma\colon\pi_pSO_{q'}\otimes\pi_qSO_{p'}\to\pi_{p+q}SO_{p'+q'-1}$$

is given by

$$d_*(\sigma(\xi_1,\xi_2))=\partial\bigl(h(\xi_1)\ast h(\xi_2)\bigr)+\overline\xi_{1*}(\xi_2)+\overline\xi_{2*}(\xi_1).$$

The formula is presented as a conjectural description of the derivative map in purely homotopy-theoretic terms; the supplied text gives no resolution status.