Which are the two conditions necessary to claim that the future spot price will have as many chances to be above or below the current forward price?
I'm not expert on this field so may not able to answer your question precisely, but I can try the best to offer you some hints.
According to the pure expectations hypothesis(PEH), forward rates provide unbiased predictions about future spot rates. Even if the PEH can be rejected, various scholars including Fama has provided evidence for the weaker form of this hypothesis. Stephen has also argued that forward rate adjusted for term-premium can be used as a predictor for future spot rate.
Francisco de Castro and Alfonso Novales has also found a robust cointegration relation between forward and future spot rate in the currency market with a unit coefficient, confirms the unbiasedness of the hypothesis. However, he also finds unbiasedness does not imply that the daily forward price is good predictor for future spot rate, as in fact, the opposite is true. This unpredictability suggests a non-stationary risk premium equal to the forward premium, making the implied forward rates to be constantly fluctuating around the true value. Therefore, if the conintegration relation is true and the conclusion of unpredictability is verified, then there is some support for your claim.