Restricted direct product is locally inner automorphism-balanced in unrestricted direct product

Statement
Suppose $$G_i, i \in I$$ is a collection of groups. Let $$G$$ be the external direct product $$\prod_{i \in I} G_i$$ and let $$H$$ be the subgroup of $$G$$ defined as the restricted direct product of the $$G_i$$s. Then, $$H$$ is a locally inner automorphism-balanced subgroup of $$G$$. In other words, for any $$g \in G$$, the inner automorphism $$x \mapsto gxg^{-1}$$ restricts to a locally inner automorphism of $$H$$: for any finite subset $$T$$ of $$H$$, there exists $$h \in H$$ such that $$gxg^{-1} = hxh^{-1}$$ for all $$x \in T$$.