Noether–Lefschetz generation conjecture for the Picard group of K3 moduli

Let $\mathcal{M}_l$ be the moduli space of quasi-polarized $K3$ surfaces, and let

$$\operatorname{Pic}(\mathcal{M}_l)^{NL}\otimes\mathbb{Q}\subset \operatorname{Pic}(\mathcal{M}_l)\otimes\mathbb{Q}$$

be the subspace spanned by the Noether–Lefschetz divisors $D_{h,d}$. Noether–Lefschetz generation conjecture. The inclusion is an isomorphism:

$$\operatorname{Pic}(\mathcal{M}_l)^{NL}\otimes\mathbb{Q}\cong \operatorname{Pic}(\mathcal{M}_l)\otimes\mathbb{Q}.$$

The conjecture concerns whether all rational divisor classes on the moduli space arise from Noether–Lefschetz divisors. The supplied source gives no evidence of a resolution, so the conjecture remains open here.