Nine-weight conjecture for the subfield code associated with the second oval polynomial

Let $q=2^m$, with $m\equiv 3\pmod 4$ and $m\geq 5$, and define

$$f(x)=x^{2^{(m+1)/2}+2^{(m+1)/4}}.$$

Let ${\mathcal{C}}_{(f,q)}^{(2)}$ be the associated binary subfield code. Nine-weight conjecture. The code ${\mathcal{C}}_{(f,q)}^{(2)}$ 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 claim is one of the conjectures prompted by Magma experiments because the minimal distance and weight distribution were not determined theoretically.