This paper explores the optimal management of defined contribution pension plans under multiple risks and strategic interactions using a stochastic differential game between two managers, accounting for model misspecification. Managers who are both competitive and averse to ambiguity can invest in a financial market that contains a risk-free asset, an inflation-linked bond, a market index, and a stock. These managers may have varying levels of aversion to ambiguous risks. The inflation index is computed using the Fisher equation. The market index and stock prices are described by a class of non-Markovian multivariate stochastic covariance models, with the market risk prices being reliant on a multivariate affine-diffusion factor process. Each manager aims to maximize the expected utility of inflation-adjusted terminal wealth relative to their competitors under the worst-case scenario of different measures, ensuring that the investment strategy is robust to model uncertainty. The robust Nash equilibrium investment strategies' explicit expressions, density generator processes under the well-defined worst-case scenarios, and corresponding value functions are derived using a backward stochastic differential equation approach to address this robust non-Markovian game. The admissibility of the robust equilibrium policies is confirmed under certain technical conditions. Finally, we provide numerical examples to demonstrate the effects of model parameters on robust investment policies and clarify the economic significance of our theoretical findings.