Generalised row-removal homomorphism conjecture for graded Specht modules

Let $\lambda,\mu\in\mathcal{P}^{l}_{n}$, $r\geq 0$, and $1\leq m\leq n$. Define $\lambda_{\mathrm{T}},\lambda_{\mathrm{B}},\mu_{\mathrm{T}},\mu_{\mathrm{B}},n_{\mathrm{T}},n_{\mathrm{B}},\mathscr{H}_{\mathrm{T}},\mathscr{H}_{\mathrm{B}}$ as above, and assume $|\mu_{\mathrm{T}}|=n_{\mathrm{T}}$. Suppose

$$\phi_{\mathrm{T}}\in\operatorname{DHom}_{\mathscr{H}_{\mathrm{T}}}(\operatorname{S}_{\lambda_{\mathrm{T}}},\operatorname{S}_{\mu_{\mathrm{T}}}),\qquad \phi_{\mathrm{B}}\in\operatorname{DHom}_{\mathscr{H}_{\mathrm{B}}}(\operatorname{S}_{\lambda_{\mathrm{B}}},\operatorname{S}_{\mu_{\mathrm{B}}}).$$

Write

$$\phi_{\mathrm{B}}(z_{\lambda_{\mathrm{B}}})=\sum_{\mathclap{t\in\operatorname{Std}_{\lambda_{\mathrm{B}}}(\mu_{\mathrm{B}})}}a_t f_t,\qquad \phi_{\mathrm{T}}(z_{\lambda_{\mathrm{T}}})=\sum_{\mathclap{s\in\operatorname{Std}_{\lambda_{\mathrm{T}}}(\mu_{\mathrm{T}})}}b_s f_s,$$

with $a_t,b_s\in\mathbb{F}$. Generalised row-removal homomorphism conjecture. There is an $\mathscr{H}_n$-homomorphism $\phi_{\mathrm{B}}\bar{\#}\phi_{\mathrm{T}}:\operatorname{S}_{\lambda}\to\operatorname{S}_{\mu}$ satisfying

$$(\phi_{\mathrm{B}}\bar{\#}\phi_{\mathrm{T}})(z_{\lambda})= \sum_{\mathclap{\substack{t\in\operatorname{Std}_{\lambda_{\mathrm{B}}}(\mu_{\mathrm{B}})\\ s\in\operatorname{Std}_{\lambda_{\mathrm{T}}}(\mu_{\mathrm{T}})}}}a_t b_s f_{t\bar{\#}s}.$$

This would give an explicit construction of the generalised row-removal isomorphism for homomorphisms between graded Specht modules, extending the preceding standard-tableau construction. The notation and the asserted construction depend on the definitions established earlier in the paper.