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Click-through rate (CTR) and post-click conversion rate (CVR) predictions are two fundamental modules in industrial ranking systems such as recommender systems, advertising, and search engines. Since CVR involves much fewer samples than CTR (known as the CVR data sparsity problem), most of the existing works try to leverage CTR&CVR multi-task learning to improve CVR performance. However, typical coarse-grained sub-network/layer sharing methods may introduce conflicts and lead to performance degradation, since not every neuron or neuron connection in one layer should be shared between CVR and CTR tasks. This is because users may have different fine-grained content feature preferences between deep consumption and click behavior, represented by CVR and CTR, respectively. To address this sharing&conflict problem, we propose a novel multi-task CVR modeling scheme with neuron-connection level sharing named NCS4CVR, which can automatically and flexibly learn which neuron weights are shared or not shared without artificial experience. Compared with previous layer-level sharing methods, this is the first time that a fine-grained CTR&CVR sharing method at the neuron connection level is proposed, which is a research paradigm shift in the sharing level. Both offline and online experiments demonstrate that our method outperforms both the single-task model and the layer-level sharing model. Our proposed method has now been successfully deployed in an industry video recommender system serving major traffic.