A more general Pandora rule?

In a model introduced by Weitzman an agent called Pandora opens boxes sequentially, in whatever order she likes, discovers prizes within, and optimally stops. Her aim is to maximize the expected value of the greatest discovered prize, minus the costs of opening the boxes. The solution, using the so-called Pandora rule, is attractive and has many applications. However, it does not address applications in which the payoff depends on all discovered prizes, rather than just the best of them, nor is it easy to say whether or not some generalized Pandora rule might do so. Here, we establish a sense in which it cannot. We discover that if a generalized Pandora rule is to be optimal for some more general utility, and all model parameters, then the problem can be solved via a second problem having Weitzman's form of utility.