In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 by Andrei Okounkov and Grigori Olshanski it is said that one of the motivations for that paper was the asymptotic character theory for the unitary groups $U(n)$ and symmetric groups $S(n)$ when $n\to\infty$ (this was first done by Vershik and Kerov). It is also said the detailed exposition of the main theorem of Vershik and Kerov regarding the characters of $U(\infty)$ via machinery of shifted Schur functions will be presented in the subsequent papers (see the part 4 of the introduction). However, I cannot find the exact paper (and not sure if it actually exists). 

Thus, the question is: did Okounkov and/or Olshanski publish somewhere a paper with the proof of the Vershik-Kerov theorem that uses their $s^*$-functions?

Here is the paper of Vershik and Kerov that Okounkov and Olshanski were referring to:

<cite authors="Vershik, A. M.; Kerov, S. V.">_Vershik, A. M.; Kerov, S. V._, Characters and factor representations of the infinite unitary group, Sov. Math., Dokl. 26, 570-574 (1982); translation from Dokl. Akad. Nauk SSSR 267, 272-276 (1982). [ZBL0524.22017](https://zbmath.org/?q=an:0524.22017).</cite>