Conjecture on reduced components of the degree-3 center variety

Let the center variety be the parameter space of centers of degree-$3$ polynomial differential equations, and consider its reduced irreducible components, classified by codimension. The conjecture specifies the following component counts:

Component-count conjecture. The number of reduced components of the center variety in degree $3$ is $1$ in codimension $5$, $2$ in codimension $6$, $4$ in codimension $7$, $4$ in codimension $8$, at least $12$ in codimension $9$, at least $33$ in codimension $10$, at least $74$ in codimension $11$, and there may be further components in codimension $12$.

These values are based on finite-field and random-point heuristics, together with comparisons against known families. In codimensions $9$ through $11$, the heuristic predicts many more reduced components than those previously identified, while codimension $12$ may contain additional components.