The third and fourth largest coset leader conjecture for odd-characteristic BCH codes

Let $q$ be an odd prime power and let $n=q^m+1$, with the quantities $\delta_3$ and $\delta_4$ defined as in the preceding cases for even $m$. Third and fourth largest coset leader conjecture. If $\delta_3$ and $\delta_4$ are given as above, then $\delta_3$ and $\delta_4$ are the third and fourth largest coset leaders, respectively. These formulas are conjectured to hold uniformly based on the Magma computations described in the paper; the statement concerns the ordering of the largest $q$-cyclotomic coset leaders relevant to BCH codes, and no proof or resolution is supplied here.