Unipotent groups with trivial $\mathbb{L}$-packets are easy

In 2006, Boyarchenko and Drinfeld conjectured that for a unipotent algebraic group over a field of positive characteristic, every geometric point is contained in the neutral connected component of its centralizer if and only if its $\mathbb{L}$-packets of character sheaves are singletons. In 2013, Boyarchenko proved the "only if" direction for $\overline{\mathbb{F}}_q$. In this paper, we complete the proof of the conjecture in this case. Along the way, we explore the relationship between general algebraic groups satisfying this property and their Asai twisting operator.