Summary: \textit{K. Takeuchi} [Asymptotic theory of statistical inference. (1974)], \textit{K. Takeuchi} and \textit{M. Akahira} [Lect. Notes Math. 550, 604-638 (1976; Zbl 0398.62027)] and \textit{J. Pfanzagl} [Contributions to Statistics, Jaroslav Hajek Mem. Vol., 167-196 (1979; Zbl 0427.62016)], among others, proved that first-order efficient estimators are second-order efficient. Many other authors have conjectured that third-order efficient estimators are also fourth-order efficient. Based on the concentration probability of estimators about a true parameter, this paper gives a positive answer to this conjecture in a curved exponential family with multi-structural parameters. It is seen that choice of bias-correction factors is critical.