Ongie et al.'s finite-representation-cost conjecture for continuous piecewise linear functions

Let $f$ be a continuous piecewise linear function, and consider shallow ReLU neural networks, allowing either finite width or infinite width with a representation cost. Ongie et al.'s conjecture. The function $f$ has finite representation cost if and only if it is exactly representable by a finite width shallow neural network. This conjecture asks whether infinite-width shallow ReLU networks provide no additional finite-cost representations among continuous piecewise linear functions beyond those already obtainable with finite width. Many compactly supported piecewise linear functions are known not to be representable with finite cost, but the stated equivalence is unresolved in the supplied source.