Ein–Erman–Lazarsfeld's normal distribution conjecture for syzygy Betti numbers

Let $X$ be a smooth projective variety of dimension $n$, let $B$ be a line bundle, and for sufficiently large $d$ set $L_d:=\mathscr{O}_X(dA+P)$. Write $k_{p,q}(X,B;L_d):=\dim K_{p,q}(X,B;L_d)$ for the Betti numbers, and let $r_d:=h^0(X,L_d)-1$. Ein–Erman–Lazarsfeld's normal distribution conjecture. For each $1\leq q\leq n$, the normalized Betti numbers should converge to a normal distribution; specifically, for $\mathbb{P}^n$ and $\mathscr{O}_{\mathbb{P}^n}(d)$, there is a normalizing function $F_q(d)$ such that

$$F_q(d)\,k_{p_d,q}(\mathbb{P}^n,\mathscr{O}_{\mathbb{P}^n}(d))\longrightarrow e^{-a^2/2}$$

as $d\to\infty$ and $p_d\to r_d/2+a\sqrt{r_d}/2$. The conjecture has not been verified even for $\mathbb{P}^2$ or $\mathbb{P}^1\times\mathbb{P}^1$, and remains challenging for Veronese embeddings.