Central-value weighting conjecture for low-lying zeros

Let $\mathcal F=\bigcup_{k=1}^{\infty}\mathcal F_k$ be a family of indices of $L$-functions, with $\#\mathcal F_k<\infty$ for each $k\geqslant 1$ and $\lim_{k\rightarrow\infty}\#\mathcal F_k=\infty$. Let $L(s,\Pi)$ be the corresponding $L$-function for $\Pi\in\mathcal F$, and suppose its one-level density satisfies

$$\lim_{k\rightarrow\infty}\frac{1}{\#\mathcal F_k}\sum_{\Pi\in\mathcal F_k}D(\Pi,\phi)=\int_{\mathbb R}\phi(x)W(\mathcal F)(x)\,dx$$

for Paley–Wiener functions $\phi$. Let $w:\mathcal F\rightarrow\mathbb C$ satisfy $\sum_{\Pi\in\mathcal F_k}w_\Pi\neq 0$ for any sufficiently large $k$, and suppose the weighted one-level density satisfies

$$\lim_{k\rightarrow\infty}\frac{1}{\sum_{\Pi\in\mathcal F_k}w_\Pi}\sum_{\Pi\in\mathcal F_k}w_\Pi D(\Pi,\phi)=\int_{\mathbb R}\phi(x)W_w(\mathcal F)(x)\,dx.$$

Here $D(\Pi,\phi)$ denotes the one-level density of low-lying zeros.

Central-value weighting conjecture. The density $W_w(\mathcal F)$ should differ from $W(\mathcal F)$ only when $w_\Pi$ essentially contains the central value $L(1/2,\Pi)$.

The conjecture proposes a general principle for weighted families of $L$-functions: central-value weights can alter the symmetry type of low-lying zeros, whereas other weights should not. The source explicitly describes this as a non-rigorous suggestion, and the paper's example shows a change for essentially central symmetric-square values when $r=2$; no general proof is given.