Unirationality conjecture for Severi varieties of unicuspidal rational curves

Let $\mathcal{V}_{\bf r}$ be the Severi variety of unicuspidal rational curves of type $({\rm S},{\bf r})$, under the hypotheses and notation of the codimension conjecture above. Unirationality conjecture. The Severi variety $\mathcal{V}_{\bf r}$ is unirational. This conjecture would give a strong birational property for the parameter spaces of unicuspidal rational curves. The supplied text gives no evidence that it has been resolved.