Anderson–Mullen conjecture on primitive and 2-primitive normal elements

Let $p\ge 5$ be a prime and let $n\ge 3$. For $\alpha\in\mathbb{F}_{p^n}$, call $\alpha$ $k$-normal over $\mathbb{F}_p$ if the $\mathbb{F}_p$-vector space spanned by

$$\{\alpha,\alpha^p,\ldots,\alpha^{p^{n-1}}\}$$

has dimension $n-k$. For $a=1,2$ and $k=0,1$, Anderson–Mullen conjecture. there exists a $k$-normal element $\alpha\in\mathbb{F}_{p^n}$ whose multiplicative order is

$$\frac{p^n-1}{a}.$$

This extends the Primitive Normal Basis Theorem by requiring simultaneous normality conditions and prescribed multiplicative orders; the source attributes the problem to Anderson and Mullen. Its resolution status is not specified in the supplied text.