Actually, prediction markets need some sort of objective, verifiable outcome, so asking "do we need a new CEO" wouldn't be appropriate. Instead the company could ask questions related to the performance metrics the CEO has agreed to meet. "Will we meet sales levels of X?" "Will we achieve x% market share in Y product by Z date?" etc.

And based on the probabilities output by the market, if they were low and trending lower, one could argue there is at least a perception that this CEO isn't getting it done. Or another response could be "why" which could encourage a more open, healthy dialogue internally.