The middle-weight conjecture for three-weight codes

Let $C$ be an $[n,(k_1,k_2)]$ $3$-weight code with dual minimum distance $d^\bot\ge 3$. Let its nonzero weights satisfy

$$w_1+w_2+w_3=3n,\qquad w_1<w_2<w_3.$$

Middle-weight conjecture. Then $w_2=n$.

The claim predicts that the middle nonzero weight of any such three-weight code equals the length. The supplied text does not indicate whether this conjecture has been proved or disproved.