The central norm criterion for homothety of Frobenius forms

Let $R$ be a Frobenius algebra with an ideal $\mathfrak m$ such that

$$R/\mathfrak m\simeq k,$$

and let $B$ and $B'$ be forms on $R$ related by $B'(r,s)=B(r,su)$ for some $u\in U(R)$. Suppose that $B$ has Nakayama automorphism $\sigma$ of finite order $n$, so that $\sigma^n=\operatorname{Id}$, and define the norm $N_\sigma(u)$ accordingly. Central norm criterion. If $N_\sigma(u)\in Z(R)$, then $B$ and $B'$ are homothetic. This is the converse to the established necessary condition that homothety implies a central norm; it extends the symmetric case $n=1$, although the source does not provide a resolution beyond this asserted converse.