Uniform boundedness of $(S{L}_2(\mathbb{C}){)}^n$ and $(PS{L}_2(\mathbb{C}){)}^n$

Let $G$ be a group and $S$ be a subset of $G.$ We say that $S$ normally generates $G$ if $G$ is the normal closure of $S$ in $G.$ In this situation, every element $g\in G$ can be written as a product of conjugates of elements of $S$ and their inverses. If $S\subseteq G$ normally generates $G,$ then the length $\Vert g{\Vert }_S\in \mathbb{N}$ of $g\in G$ with respect to $S$ is the shortest possible length of a word in ${\text{Conj}}_G(S^{\pm 1}):=\{h^{-1}sh|h\in G,s\in S\text{or}s{}^{-1}\in S\}$ expressing $g.$ We write $\Vert G{\Vert }_S=\text{sup}\{\Vert g{\Vert }_S|g\in G\}$ for any normally generating subset $S$ of $G.$ The conjugacy diameter of any group $G$ is $\mathrm{\Delta }(G):=sup\{\Vert G{\Vert }_S|S\text{is a finite normally generating subset of }G\}.$ We say that $G$ is uniformly bounded if $\mathrm{\Delta }(G)<\mathrm{\infty }.$ This concept is a strengthening of boundedness. Motivated by previously known results approximating $\mathrm{\Delta }(G)$ for any algebraic group $G,$ we find the exact values of the conjugacy diameters of the direct product of finitely many copies of $S{L}_2(\mathbb{C})$ and the direct product of finitely many copies of $PS{L}_2(\mathbb{C}).$ We also prove that if $G_1,\dots ,G_n$ be quasisimple groups such that $G_i$ is uniformly bounded for each $i\in \{1,\dots ,n\},$ then $G_1\times \cdots \times G_n$ is uniformly bounded. This is also a generalization of some previously known results in the literature.