It is well known that increase of the spatial dimensionality enhances the fluid&-fluid demixing of a binary mixture of hard hyperspheres, i.e. the demixing occurs for lower mixture size asymmetry as compared to the three-dimensional case. However, according to simulations, in the latter dimension the fluid&-fluid demixing is metastable with respect to the fluid&-solid transition. According to the results obtained from approximations to the equation of state of hard hyperspheres in higher dimensions, the fluid&-fluid demixing might become stable for high enough dimension. However, this conclusion is rather speculative since none of these works have taken into account the stability of the crystalline phase (by a minimization of a given density functional, by spinodal calculations or by MC simulations). Of course, the lack of results is justified by the difficulty of performing density functional calculations or simulations in high dimensions and, in particular, for highly asymmetric binary mixtures. In the present work, we will take advantage of a well tested theoretical tool, namely the fundamental measure density functional theory for parallel hard hypercubes (in the continuum and in the hypercubic lattice). With this, we have calculated the fluid&-fluid and fluid&-solid spinodals for different spatial dimensions. We have obtained, no matter what the dimensionality, the mixture size asymmetry or the polydispersity (included as a bimodal distribution function centered around the asymmetric edge lengths), that the fluid&-fluid critical point is always located above the fluid&-solid spinodal. In conclusion, these results point to the existence of demixing between at least one solid phase rich in large particles and one fluid phase rich in small ones, preempting a fluid&-fluid demixing, independently of the spatial dimension or the polydispersity.