A conjecture is made for the expected value of the $j$th elementary symmetric function (ESF) of the roots of a noncentral Wishart matrix with covariance matrix $\sigma^2I$. A conjecture is also made if $j = p$ for any covariance matrix $\Sigma$. The expected value of the noncentral Wishart matrix is derived for any covariance matrix $\Sigma$ and therefore also the expected value of the first ESF of the roots.
D. J. de Waal. "On the Expected Values of the Elementary Symmetric Functions of a Noncentral Wishart Matrix." Ann. Math. Statist. 43 (1) 344 - 347, February, 1972. https://doi.org/10.1214/aoms/1177692728