The set of forms with bounded strength is not closed

The strength of a homogeneous polynomial (or form) $f$ is the smallest length of an additive decomposition expressing it whose summands are reducible forms. We show that the set of forms with bounded strength is not always Zariski-closed. In particular, if the ground field has characteristic $0$, we prove that the set of quartics with strength $leq3$ is not Zariski-closed for a large number of variables.