Variance of the Wishart Distribution when $p$ is larger than $n$

Question

I came across this theorem mentioned in Srivastava (2005) https://www.jstage.jst.go.jp/article/jjss/35/2/35_2_251/_pdf/-char/ja

click here for the lemma

The paper is talking about the high-dimensional case where number of variables $p$ is larger than the sample size. I have learned from Ledoit and Wolf (2003) that under this case the sample covariance matrix isn't consistent anymore. Does this correspond to the variance of the Wishart distribution here behaving strangely under $p>n$?

Another question I have is, suppose I want to derive the distribution of some functions of the sample covariance matrix using the Delta method (say $s_{11}^2+s_{22}^2+s_{33}^2+...+s_{pp}^2$), is it still ok to use this lemma under $p>n$ case or is this incorrect?

Thank you very much for your help!