It depends on what information you’re willing to leverage to impute the missing predictors. If you’re willing to assume that predictors are drawn from some common distribution across timesteps, then in principle you can fit these distributions across timesteps, and with more data from more timesteps you could get better inference. On the other hand, if you’re just imputing the probable values of missing predictors based on the observed value of the response and the modeled relationships between the predictors and the response (i.e. you’re not incorporating any information about the probable values of the predictors except by conditioning on the response) then you don’t get any benefit.