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The redshift of $z\sim1.5$ is the transition epoch of protoclusters (PCs) from the star-forming phase into the quenching phase, and hence an appropriate era to investigate the build up of the quenched population. We define a `core' as the most massive halo in a given PC, where environmental effects are likely to work most effectively, and search for cores at $1<z<1.5$. We use a photometric redshift catalogue of a wide (effective area of $\sim22.2\,\mathrm{deg}^{2}$) and deep ($i\sim26.8\,\mathrm{mag}$) optical survey with Subaru Hyper-Suprime Cam. Regarding galaxies with $\log(M_{*}/M_{\odot})>11.3$ as the central galaxies of PC cores, we estimate their average halo mass by clustering analysis and find it to be $\log(M_\mathrm{h}/M_{\odot})\sim13.7$. An expected mass growth by the IllustrisTNG simulation and the observed overdensities around them suggest that the PC cores we find are progenitors of present-day clusters. Classifying our galaxy sample into red and blue galaxies, we calculate the stellar mass function (SMF) and the red galaxy fraction. The SMFs in the PC cores are more-top heavy than field, implying early high-mass galaxy formation and disruption of low-mass galaxies. We also find that the red fraction increases with stellar mass, consistent with stellar-mass dependent environmental quenching recently found at $z>1$. Interestingly, although the cores with red and blue centrals have similar halo masses, only those with red centrals show a significant red fraction excess compared to the field, suggesting a conformity effect. Some observational features of PC cores may imply that the conformity is caused by assembly bias.