At , galaxies in proto-clusters (PCs) are actively star forming, while matured clusters at are dominated by quenched galaxies, suggesting that is a transition epoch of (proto-)clusters from star forming phase into quenching phase. To investigate galaxy evolution in this transition epoch, we search for PC cores at . Here, a "core" is defined as the most massive halo in a given PC, where environmental effects are likely to work most effectively. Using a photo-z catalog of large and deep optical survey data with Subaru Hyper-Suprime Cam, we search for PC cores in a very wide field that reaches . Regarding galaxies more massive than as the central galaxies of PC cores, we detect more than 1000 core candidates. Then, we estimate their average halo mass by clustering analysis and find it to be . From a comparison with the Illustris TNG simulation, we confirm that these massive halos at are progenitors of present-day clusters comparable to or more massive than Fornax-type clusters. Classifying our galaxy sample into red and blue galaxies, we calculate the red fraction of the member galaxies of our PC core candidates. Although red and blue centrals have similar halo masses, only the red fraction around red centrals shows an excess compared to the field, while that around blue centrals shows no excess, suggesting a conformity effect. We discuss galaxy quenching in PC cores at .