There is no way to "establish whether or not there is an objective reality." It's a philosophical position. You either take the reality which we observe and study as part of the material sciences to be objective reality, or you don't believe it's objective reality and think it is all sort of invented in the "mind" somehow. Either position you take, you cannot prove or disprove either one, because even if you take the latter position, no evidence I present to you could change your mind because to be presented evidence would only mean for that evidence to appear in the mind, and thus wouldn't prove anything. The best argument we can make is just taking the reality we observe as indeed reality is just philosophically simpler, but that also requires you to philosophically value simplicity, which you cannot prove what philosophical principles we should value with science either.