The sine-process maximum-growth conjecture at zero shift

Let $X$ be a configuration sampled from the sine process, let $G_{X+iA}$ be the associated random entire function, and define

$$M_A(T)=\max_{t:|t|<T}\log\lvert G_{X+iA}(t+iA)\rvert.$$

For $A>0$, the known asymptotic is

$$\limsup_{T\to\infty}\frac{M_A(T)}{\log T}=\sqrt{2}.$$

The sine-process maximum-growth conjecture at zero shift. The same asymptotic formula holds for $A=0$:

$$\limsup_{T\to\infty}\frac{\max_{t:|t|<T}\log\lvert G_X(t)\rvert}{\log T}=\sqrt{2}.$$

The supplied text states the formula for positive shifts and leaves the zero-shift case as an interesting direction.