The node-existence conjecture for high-degree plane curves with nodes and triple points

Let $C\subset\mathbb{P}^2$ be a reduced plane curve whose singularities are all nodes or triple points, and let $n_2$ denote the number of nodes. Node-existence conjecture. If $C$ has degree $d>9$, or if $C$ is irreducible and has degree $d>8$, then

$$n_2>0.$$

This is presented as a variation of an earlier conjecture and as a possible guide to constructing counterexamples to the irreducible sigma-two conjecture. The source gives no resolution.