2 problems

Nonemptiness conjecture. For every form $F$, the fertile locus $\mathcal{F}_F$ is nonempty.

Waring-locus additivity conjecture. If $F=\sum_{i=1}^sF_i\in S$ is a degree $d\geq3$ form such that $F_i\in S^{[i]}$ for all $i=1,\ldots,s$, then