Normal distribution conjecture for asymptotic Betti tables

Let $X$ be a smooth projective variety of dimension $n$, let $L_d$ be the very positive line bundles used to define the Betti numbers $k_{p,q}(X;L_d)$, and write $r_d$ for the relevant embedding dimension. Fix a weight $1\leq q\leq n$ and a sequence of integers $p_d$ satisfying

$$p_d\to \frac{r_d}{2}+a\cdot\frac{\sqrt{r_d}}{2}$$

for a fixed number $a$. Asymptotic normal-distribution conjecture. There is a normalizing function $F_q(d)$, depending on $X$ and geometric data, such that

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

as $d\to\infty$ and $p_d\to \frac{r_d}{2}+a\cdot\frac{\sqrt{r_d}}{2}$. Equivalently, the rows of the Betti table of any very positive embedding should display roughly the pattern of a large Koszul complex. The conjecture extends the proved normal-distribution asymptotics for curves to arbitrary smooth projective varieties, but no resolution is supplied here.