The conjecture that the maximal order of S(t) is one half

Let $N(t)$ count the nontrivial zeros of the Riemann zeta function up to height $t$, and define $S(t)$ by

$$N(t)=\frac{t}{2\pi}\log \Big(\frac{t}{2\pi e}\Big)+\frac{7}{8}+S(t)+O(1/t).$$

Define

$$\alpha=\limsup_{t\to\infty}\frac{\log |S(t)|}{\log\log t}.$$

The maximal-order conjecture for $S(t)$. Conditionally on the Riemann hypothesis,

$$\alpha=1/2.$$

The source records the conditional bounds $1/2\leq\alpha\leq 1$ and explains that the conjecture identifies the expected true maximal order of the zero-counting remainder. The source does not state that this conjecture has been resolved.