Let $G$ be a finite $p$-group and $(\mathbb{F},\mathcal{O},\mathbb{K})$ a sufficiently large $p$-modular system. Furthermore let $\mu_{p^\infty}\leq \mathcal{O}^\times$ the group of roots of unity of $p$-power order.
If $\alpha$ is a 2-cocyle with values in $\mu_{p^\infty}$ does there always exist a one-dimensional projective representation $G\to\mathcal{O}^\times$ with cocylce $\alpha$ ? In other words: Does the twisted group algebra $\mathcal{O}_\alpha[G]$ always have a one-dimensional representation $\mathcal{O}_\alpha[G]\to\mathcal{O}$?