Necessary and sufficient conditions for the strong renewal theorem for random walks

Let $S$ be an aperiodic random walk whose increment distribution belongs to $D(\alpha,\rho)$, where $\alpha\in(0,1)$ and $\rho>0$. Let and denote the conditions stated in the source, and let denote the strong renewal theorem (SRT). Random-walk strong renewal conjecture. The conditions and are necessary and sufficient for the SRT to hold for any such random walk. The question asks whether the renewal-process result extends to random walks; the supplied text does not establish the assertion or provide evidence that it has been resolved.