Gaussian asymptotics conjecture for Betti numbers of large-degree embeddings

Let $X$ be a smooth projective variety of dimension $n$, let $A$ be a line bundle defining the large-degree embeddings, and set

$$L_d=dA+P,$$

with $r_d=h^0(X,L_d)-1$. Write

$$k_{p,q}(X;L_d)=\dim K_{p,q}(X;L_d).$$

Gaussian Betti-number conjecture. Fix $q\in[1,n]$. There is a normalizing function $F_q(d)$, depending on $X$ and geometric data, such that for every sequence of integers $p_d$ satisfying

$$p_d\longrightarrow \frac{r_d}{2}+a\frac{\sqrt{r_d}}{2},$$

for fixed $a$, one has

$$F_q(d)\,k_{p_d,q}(X;L_d)\longrightarrow e^{-a^2/2}$$

as $d\to\infty$.

The conjecture proposes that the Gaussian profile proved for weight-one syzygies of curves persists universally for all smooth projective varieties and weights. The source presents this as an expected asymptotic pattern; no resolution is given.