$\mathrm{\Theta }$-Hilbertianity and strong $\mathrm{\Theta }$-Hilbertianity

$\mathrm{\Theta }$-Hilbertianity and its strengthening, strong $\mathrm{\Theta }$-Hilbertianity, are two generalizations of Hilbertianity inspired by Jarden’s definition of $p$-Hilbertianity and strong $p$-Hilbertianity. Jarden has asked whether the two notions defined by him are actually the same. We address this question in its more general version of $\mathrm{\Theta }$-Hilbertianity and show that for PRC, and, in particular, for PAC fields, $p$-Hilbertianity and strong $p$-Hilbertianity coincide.