  In this paper we study the category of Harish-Chandra bimodules
$HC_{\chi,\psi}$ in the Deligne category $\text{Rep}(GL_t)$. In particular, we
answer Question 3.25 posed in Pavel Etingof's paper arXiv:1407.0373 and
determine for which central characters $\chi$ and $\psi$ this category is not
zero.
