  We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension
$\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive
element $\xi$ such that $\xi+\xi^{-1}$ is also primitive, and
$\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(\xi)=a$ for any prescribed
$a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma,
and Cohen concerning the analogous problem over an extension of arbitrary
degree $n\ge3$.
