Persistence exponent conjecture for fixed-size truncations of random orthogonal matrices

Let $\ell$ be a fixed integer, and let $M_{2n}$ be the ensemble of random matrices defined in Theorem 1. The corresponding persistence probability is denoted by $p_{2n}^{(\ell)}$. Define

$$\theta(\ell)=-\frac{1}{2\pi}\int_0^\infty\log\left(1-\left|\frac{\Gamma\left(\frac{\ell}{2}+ix\right)}{\Gamma\left(\frac{\ell}{2}\right)}\right|^2\right)\,\mathrm{d}x.$$

Persistence exponent conjecture. The persistence probability satisfies

$$\lim_{n\to\infty}\frac{\log p_{2n}^{(\ell)}}{\log n}=-2\theta(\ell).$$

For $\ell=1$ this asymptotic behavior is established in the paper, while the conjecture extends the persistence-exponent formula to every fixed integer $\ell>1$.