Generalized Zariski component conjecture for branch-curve strata

For $ue 3$, define the numerical invariants

$$d( u)= u( u-1),$$ $$c( u)= u( u-1)( u-2),$$ $$n( u)=\frac{1}{2}\nu(\nu-1)(\nu-2)(\nu-3).$$

Let $V(d,c,n)$ be the space of irreducible plane curves of degree $d$ with $c$ cusps and $n$ nodes as their only singularities, and let $B(d,c,n)$ be the family of branch curves of generic projections of smooth surfaces of degree $u$ in $cmathbb{P}^3$. Generalized Zariski component conjecture. For each $u\ge 3$, the variety $V(d( u),c( u),n( u))$ contains a component disjoint from $B(d( u),c( u),n( u))$. This generalizes the known degree-$3$ case, where $V(6,6,0)$ has a component disjoint from $B(6,6,0)$, and predicts such non-branch components in every degree $u\ge 3$.