A Counterexample to Wehlau's Conjecture on Noether Numbers

Let $G$ be a finite group, let $V$ be a finite-dimensional $G$-module over a field $k$, and let $U$ be a $G$-submodule of $V$. Wehlau conjectured that the corresponding Noether numbers satisfy $\beta\bigl(k[U]^G\bigr)\leq \beta\bigl(k[V]^G\bigr)$. We disprove this conjecture in characteristic $2$. For the dihedral group $D_8$, we construct an inclusion $U\subseteq V$, with $\dim_kU=5$ and $\dim_kV=6$, such that $\beta\bigl(k[U]^{D_8}\bigr)=6>5=\beta\bigl(k[V]^{D_8}\bigr)$. The construction is defined over $\mathbb F_2$ and remains valid over every field of characteristic $2$. By inflation, it yields counterexamples for every finite group admitting $D_8$ as a quotient.