Does $\sum_n |\sin n|^{cn^2}$ converge?

So I recently asked a question about convergence of $\sum_n |\sin n|^{cn}$ for arbitrary $c > 0$ and it turns out that the terms of the series don’t even converge, for any $c > 0$, so the series is always divergent. But what about $\sum_n |\sin n|^{cn^2}$ for $c > 0$? Are there $c$ so that the series converges, and if there are $c > 0$ such that the series diverges, do the terms of the series still converge?

If that’s too hard, what if we replace $n^2$ in the exponent by $n^\alpha$ for some different $\alpha > 1$? Which $\alpha$ do we know the answer for?

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