In the ring of shifted symmetric functions $\Lambda^*$ there are many ways to generalize the symmetric power sums. First of all, we have the functions $$p^*_k=\sum_{i=1} \left((x_i-i+1/2)^k-(-i+1/2)^k\right)$$ (the factor 1/2 is convenient when the $x_i$ are the parts of a partition, but can replaced by any constant). For a partition $\kappa$ we let $p^*_\kappa=p^*_{\kappa_1}p^*_{\kappa_2}\cdots.$ Another shifted analogue to the power sums are the functions $p_\kappa^\#$, defined by $$p^\#_\kappa = \sum_{\lambda \vdash k} \chi_\kappa^\lambda s_\lambda^*,$$ where k=$|\kappa|$ is the size of the partition $\kappa$. Both $p_\kappa^*$ and $p_\kappa^\#$ provide a basis for $\Lambda^*$. (For more details: see the original paper 'Shifted Schur Functions' by Okounkov and Olshanski or the post On shifted symmetric power sums).

In the literature there is a vast amount of results relating different bases of $\Lambda^*$ to each other. However, the only relation between the bases by $p_\kappa^*$ and $p_\kappa^\#$ I can find is that $$p_\kappa^\#=p_\kappa^*+\textrm{lower degree terms},$$ (because the highest degree terms of $p_\kappa^\#$ and $p_\kappa^*$ equal the ordinary symmetric power sum $p_\kappa$). How can these lower degree terms be expressed in terms of $p_\kappa^*$ (or $p_\kappa^\#$)?

Question: Are there any expressions known relating $p_\kappa^*$ and $p_\kappa^\#$?