Lower degree bounds for modular vector invariants

Let $G$ be a finite group of order divisible by a prime $p$ acting on an $\mathbb{F}$ vector space $V,$ where $\mathbb{F}$ is the field with $p$ elements and $\dim_{\mathbb{F}} V=n$. Consider the diagonal action of $G$ on $m$ copies of $V.$ This note sharpens a lower bound for $\beta(\mathbb{F}[\oplus_mV]^G)$ for groups which have an element of order $p$ whose Jordan blocks have sizes at most 2.