For all $j \in \mathbb{N}$, does there exist $k \in \mathbb{N}$ such that $\{j^{k+1} \pi^k/2\} > 1/2$?

The question is in the title: if you pick a natural number $j$, are you always guaranteed to have the fractional part of some $j^{k+1} \pi^k/2$ be strictly greater than $1/2$? One would certainly expect this to be true from normality/equidistribution-style properties, though those are notoriously difficult/unproven. Nonetheless, this is about the weakest form of such a property you could hope for, so I wonder if more can be said.

The motivation is this question, where one is interested in $\alpha > 1$ for which $\sin(\alpha^n) > 0$ for all $n \in \mathbb{N}$. The argument listed there proves there is a smallest such $\alpha$ provided the requirement is weakened slightly to $\sin(\alpha^n) \geq 0$. But it seems terribly unlikely the two statements are actually different. If the above property holds, then they're the same.