Unboundedness conjecture for the prime-index partial sums

Let $x\geq 1$ and define the prime-index function by $\iota(n)=(-1)^{\pi(n)}$, where $\pi(n)$ is the prime-counting function. Define its partial sums by

$$\xi(x)=\sum_{n\leq x}\iota(n).$$

Unboundedness conjecture. For every $N\geq N_0$, where $N_0>0$ is sufficiently large, the supremum of these partial sums satisfies

$$\sup_{x\geq 1}\left\{\sum_{n\leq x}(-1)^{\pi(n)}\right\}\geq N.$$

The claim asserts that the prime-index partial sums are unbounded above. The source presents it as a belief motivated by the observed distribution of $\xi(x)$, but gives no resolution.