The Ramanujan nearest-integer conjecture for the prime-counting function

Let $\mathcal{R}\subset \mathbb{R}_+$ be the Ramanujan set of values $x\in \mathbb{R}_+$ for which $\pi(x)$ is the nearest integer to

$$\sum_{n=1}^{+\infty} \frac{\mu(n)}{n}\ \mathrm{li}(x^{\frac{1}{n}}) \,,$$

where $\pi(x)$ is the prime-counting function, $\mu$ is the Möbius function, and $\mathrm{li}$ is the logarithmic integral. Ramanujan's conjecture. The set $\mathcal{R}$ contains arbitrarily large numbers. This interprets Ramanujan's claim that his asymptotic formula is generally equal to the prime-counting function up to the nearest integer. The source does not give evidence of a resolution.