Conjecture on generators with nonsquare quadratic translate over finite fields

Let $p$ be an odd prime, and write $(\mathbb{Z}/p\mathbb{Z})^{*}$ for the multiplicative group of nonzero residue classes modulo $p$. A residue class $\overline{k}$ is a generator if it generates this group, and a residue class is a square modulo $p$ if it is a square in $(\mathbb{Z}/p\mathbb{Z})^{*}$. The generator–nonsquare conjecture. For every odd prime $p$, there exists a generator $\overline{k}$ of $(\mathbb{Z}/p\mathbb{Z})^{*}$ such that $\overline{k}^{2}+\overline{4}$ is not a square modulo $p$. The conjecture was verified computationally for all primes between $3$ and $2\,000\,000$, but no general proof is given.