Exceptional sine-sum conjecture

Let $q$ and $n$ be positive integers, and let $p_1,\ldots,p_n$ be distinct integers relatively prime to $q$ such that $q>(7/4)^n$ and

$$\sum_{k=1}^n p_k\leq q.$$

Assume that for every $k\in\{1,\ldots,n\}$,

$$\frac{2}{\sin(\pi/q)}\leq\sum_{i=1}^n\frac{1}{\left|\sin(\pi p_k\bar p_i/q)\right|},$$

where $\bar p_i$ denotes the modular inverse of $p_i$ modulo $q$. Exceptional sine-sum conjecture. Then

$$q=2^n-1$$

and

$$\{p_1,\ldots,p_n\}\equiv\{1,2,\ldots,2^{n-1}\}\pmod q.$$

This conjecture is used to sharpen an exponential bound in the attempted proof of Fraenkel's conjecture. The paper reports computational support but does not establish the assertion.