Abstract This paper explores implications of weak identification in common ‘long memory’ and recent ‘rough’ approaches to modeling volatility dynamics of financial assets. We unveil an asymptotic near-observational equivalence between a long memory model with weak autoregressive dynamics and a rough model with a near-unit autoregressive root. Standard methods struggle to distinguish them, and conventional asymptotics are invalid. We propose an identification-robust approach to construct confidence sets that reveal the uncertainty and aid inference. Empirical studies based on realized volatility and trading volume often fail to statistically reject either model, thereby providing evidence of their potential coexistence.