The degree bound conjecture for field invariants of cyclic prime-group representations

Let $G=\mathbb{Z}/p\mathbb{Z}$ with $p$ an odd prime, let $\Bbbk$ be a field of characteristic different from $p$, and let $V$ be a representation of $G$. Let $m$ be the number of distinct, nontrivial isotypic components in the base change of $V$ to an algebraic closure $\overline{\Bbbk}$. Degree bound conjecture. The field Noether bound satisfies

$$\beta_{\mathrm{field}} \leq \left\lceil\frac{p}{\lceil m/2\rceil}\right\rceil.$$

The preceding general upper bound is not sharp, and computational data suggest this sharper estimate; the source also indicates that, if correct, it is sharp even for the corresponding robust invariant bound.