Nine-weight conjecture for the subfield code associated with $x^6$

Let $q=2^m$, and let ${\mathcal{C}}_{(f,q)}^{(2)}$ be the binary subfield code defined by the trace representation for an oval polynomial $f$. Nine-weight conjecture. For odd $m\geq 5$, the code ${\mathcal{C}}_{(x^6,q)}^{(2)}$ has parameters

$$[2^m+1,2m,2^{m-1}-2^{(m-1)/2}],$$

and nine nonzero weights. Its dual has parameters $[2^m+1,2^m-2m+1,3]$. The minimal distance and weight distribution were not determined in the paper, and this claim is based on Magma experiments.