Strong radical Sylvester–Gallai conjecture

Let $\rK$ be the base field, let $d\in\N$, and let $\cF$ be an $\rsg{d}$ configuration. Write $\Kspan{\cF}$ for its $\rK$-linear span. Strong radical Sylvester–Gallai conjecture. There is a function

$$\lambda: \N \to \N$$

such that

$$\dim\bigl(\Kspan{\cF}\bigr)\leq\lambda(d)$$

for every $\rsg{d}$ configuration $\cF$. This is explicitly presented as a stronger version of the radical Sylvester–Gallai conjecture, since transcendence degree is bounded by the dimension of the linear span; the source supplies no resolution, so the conjecture remains open.