What you are stating is the original Lottery Ticket Hypothesis. What they prove in this paper is the stronger version, empirically noticed here https://arxiv.org/abs/1905.01067 and referred to as "supermasks". To quote from the paper posted here: "within a sufficiently overparameterized neural network with random weights (e.g. at initialization), there exists a subnetwork that achieves competitive accuracy".

Edit: See also https://arxiv.org/abs/1911.13299