Minimum-weight codeword conjecture for degree-4 maximal arcs

Let $D$ be a design associated with a maximal arc of degree $4$ in $PG(2,2^m)$, where $m\ge 2$, and let $A$ be its incidence matrix. Let $C$ be the binary code spanned by the rows of $A$. Minimum-weight codeword conjecture. The code $C$ has minimum distance $4$, and every codeword of minimum weight is a row of $A$. This conjecture arises from computations with resolvable $2$-$(52,4,1)$ designs associated with maximal arcs in projective planes of order $16$; its validity in general is not established by the supplied text.