Suppose $\lambda\vdash n$ is a partition and $S^\lambda$ is the associated irreducible representation of $S_n$. As we know from the branching rule, we have an isomorphism of $S_{n-1}$ modules $$S^\lambda\cong\bigoplus_{\mu}S^{\mu}$$ where $\mu$ are the partitions obtained by deleting an outer corner from $\lambda$.

In particular, I am concerned with the case when $S^\lambda$ is spanned by basis elements $\{B_Q\mid Q\in\text{SYT}(\lambda)\}$ of the Kazhdan-Lusztig basis (with the relevant left-action of the generators). Can the isomorphism from the branching rule be realised in such a way that preserves the KL-basis in some form? That is, if $\phi$ is a realisation of the isomorphism, then $$\phi\cdot B_Q=\sum_{\mu}\sum_{P\in\text{SYT}(\mu)}z_pB_P$$ for some constants $z_P$. What can we say about these constants $z_P$?

For example, when $\lambda$ is a rectangular partition (and so $S^\lambda\cong S^\mu$), then the isomorphism can be realised as $B_Q\mapsto B_{d(Q)}$, where $d(Q)$ is the tableau obtained by deleting the $n$-box from $Q$. I am unsure as to the theory for arbitrary partitions.

Any help would be appreciated!