The pseudo-hyperjump operator is a generalization of the classical hyperjump, mapping Turing degrees to higher degrees in a structured way. The paper establishes that full inversion fails for Turing degrees in general, meaning not every degree above a certain threshold is realized as a pseudo-hyperjump of some degree. The open problem is whether one can identify a natural, well-defined class of Turing degrees, somewhere in the vast territory between the arithmetical degrees and the hyperdegrees, for which pseudo-hyperjump inversion does hold, and whether such a class can be characterized by purely degree-theoretic or definability-theoretic properties rather than by ad hoc construction.