The limsup conjecture for the zeta zero-counting error term

Let $S(t)$ denote the error term in the number of zeros of the Riemann zeta-function up to height $t$. Limsup conjecture.

$$\limsup_{t\to\infty}\frac{S(t)}{\sqrt{\log t\log\log t}}=\frac{1}{\pi\sqrt{2}}.$$

The conjecture is motivated by arguments analogous to those used for large values of the zeta-function and predicts the precise leading constant for the extreme size of $S(t)$.