Twisted Convolution Conjecture (shifts and convolution identity)
Prove that for every admissible tuple t, the set of allowed shifts Z_t contains 0 and 1, and that Z_t depends only on the discriminant of the associated form. Equivalently, establish the twisted convolution identity stipulated in Definition 1.XX for all p with any such shift.
References
We are now ready to state our additional conjecture: \begin{conj}[Twisted Convolution Conjecture]\label{cnj:tci} For every admissible tuple the set of shifts $\mcl{Z}t$ includes the values $\lambda = 0, 1$. Moreover, if $t=(d,r,Q)$ and $t'=(d,r,Q')$ are admissible tuples such that $Q$ and $Q'$ have the same discriminant, then $\mcl{Z}_t = \mcl{Z}{t'}$. \end{conj}
— A Constructive Approach to Zauner's Conjecture via the Stark Conjectures
(Appleby et al., 7 Jan 2025) in Subsection 1.6 (The main conjectures)