Property $P_{\text{naive}}$ for big mapping class groups

We study the property $P_{\text{naive }}$ of mapping class groups of surfaces of infinite type, that is, for any finite collection of non-trivial elements $h_1,h_2,\cdots ,h_n$, there exists another element $g\mathrm{\ne }1$ of infinite order such that for all $i$, $⟨g,h_i⟩\cong ⟨g⟩\ast ⟨h_i⟩$.