Universal generator-set conjecture for circulant nut graphs

For a finite set $S$ of positive integers, call $S$ universal when $\operatorname{Circ}(n,S)$ is a nut graph for every even $n\geq 2\max(S)+2$. Here $\operatorname{Circ}(n,S)$ is the circulant graph with vertex set $\{0,1,\dots,n-1\}$ and edges $(i,i\pm s)$ modulo $n$ for $s\in S$.

Universal generator-set conjecture. For each odd $t\geq 3$, there exists $p_t$ such that $\operatorname{Circ}(n,\{1,\dots,2t+1\}\setminus\{p_t\})$ is a nut graph for every even $n\geq 4t+4$. For each even $t\geq 4$, there exist $q_t$ and $r_t$ such that $\operatorname{Circ}(n,\{1,\dots,2t+2\}\setminus\{q_t,r_t\})$ is a nut graph for every even $n\geq 4t+6$.

The conjecture extends the theorem covering many odd values of $t$ and is motivated by computational evidence through $t=1300$ for the remaining odd cases and for even $t$; a general proof is not supplied.