Pre-RdLi-commutativity implies strong RdLi-commutativity

Let $\sigma, \sigma' in \mathrm{Irr}^{square}$ and let $pi in \mathrm{Irr}$. A triple $(\sigma,\sigma',pi)$ is pre-RdLi-commutative when it satisfies the pre-RdLi-commutativity condition, and it is strongly RdLi-commutative when the associated map factors through the specified derivative embedding. Pre-RdLi commutativity conjecture. If $(\sigma,\sigma',pi)$ is pre-RdLi-commutative, then it is strongly RdLi-commutative. This conjecture would identify the pre- and strong commutativity conditions in the setting relevant to the paper's duality results; no resolution is supplied in the source.