Sun’s circular-permutation conjecture over finite fields

Establish whether, for every prime power q>7 and every a_0\in\mathbb{F}_q, there exists a circular permutation (a_1,\dots,a_{q-1}) of all elements of \mathbb{F}_q^* such that each of a_0+a_1a_2, a_0+a_2a_3, \dots, a_0+a_{q-2}a_{q-1}, and a_0+a_{q-1}a_1 is a primitive element of \mathbb{F}_q, including the cases not covered by the paper’s proof for q>18,888,871.

Background

The paper reformulates Sun’s conjecture as a Hamiltonicity problem for the graph G_{q,a_0} whose vertex set is \mathbb{F}_q^* and in which distinct vertices x and y are adjacent precisely when a_0+xy is primitive. A Hamilton cycle in this graph gives the required circular permutation.

The paper proves the conjecture for every prime power q>18,888,871 and proves the a_0=0 case for q>4. Consequently, the general conjecture as stated remains unresolved for the parameter range not covered by those results, particularly 7<q\le18,888,871 with a_0\ne0.