Weak cosmic censorship conjecture in Einstein-Maxwell gravity with scalar hair

In this paper, we focus on the weak cosmic censorship conjecture (WCCC) in the Einstein-Maxwell gravity sourced by a conformally coupled scalar field, where the black hole solution can carry a scalar hair apart from the conserved charges of the electromagnetic field. Motivated by the new version of the gedanken experiments proposed by Sorce and Wald, we derive the first two order perturbation inequalities in the scalar-hair Reissner-Nordstrom (shRN) black holes based on the Iyer-Wald formalism under the stability consumption. In the collision process, we assume that the scalar hair can also be changed. As a result, we find that the nearly extremal shRN black holes cannot be destroyed by the above collision process under the second-order approximation after the two perturbation inequalities are considered. Our result at some level implies that the WCCC can also be restored in the black holes with some scalar hairs.