Semi-algebraicity conjecture for the maximal honest open in ReLU-network parameter space

For a fixed ReLU-network architecture, let $\mathcal{P}$ denote the parameter space obtained from the space of weights by taking into account the trivial scaling and permutation symmetries. An open subset of $\mathcal{P}$ is honest when the parametrization has no hidden symmetries over it, and strongly honest when the corresponding symmetry action is also free. Semi-algebraicity conjecture. The maximal honest open, respectively the maximal strongly honest open, for a ReLU neural network of fixed architecture is semi-algebraic. Such opens describe the regions on which the network is identifiable up to scaling and permutations. The source notes that prior work gives generic or positive-measure identifiability and implies the existence of weakly honest semi-algebraic opens, while the semi-algebraicity of the maximal honest and strongly honest opens is not established.