Bazzi–Mitter conjecture on hyperelliptic-curve point bounds

For an odd prime $p$ and each non-empty subset $S\subset GF(p)$, let $X_S$ be the hyperelliptic curve defined by

$$y^2=f_S(x),\qquad f_S(x)=\prod_{a\in S}(x-a).$$

Write $B(c,p)$ for the assertion that $|X_S(GF(p))|\leq c\,p$ for every subset $S\subset GF(p)$. Bazzi–Mitter conjecture. There is a $c\in(0,2)$ such that $B(c,p)$ holds for infinitely many primes $p$. This conjecture seeks a uniform improvement over the trivial bound $B(2,p)$ and is connected in the source to constructing binary codes with positive relative distance; the source gives no resolution.