On club-like principles on regular cardinals above $\beth_\omega$

We prove that for regular $\lambda$ above a strong limit singular $\mu$ certain guessing principles follow just from cardinal arithmetic assumptions. The main result is that for such $\lambda$ and $\mu$ there are coboundedly many regular $\kappa<\mu$ such that $\clubsuit^-(S^\lambda_\kappa)$ holds whenever $\lambda=\lambda^{<\kappa}$.