Let $U_\infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_\infty$. That is, $U_r \subseteq U_{r'}$ if $r \le r'$ and $U_\infty = \overline{\bigcup U_r}$.
For $r \in [1,\infty]$, let $Y_r = C(U_r,\mathbb R)$ be the Banach space of real-valued continuous functions over $U_r$ with the supremum norm. For $r \le r'$, let $\phi_{r,r'} : Y_{r'} \to Y_r$ be the restriction maps, so that $Y_\infty$ is the inverse limit of the spaces $Y_r$. Write $\phi_r : Y_\infty \to Y_r$ for the restriction map $\phi_{r,\infty}$.
Suppose there exists a family of continuous linear operators $m_r : Y_r \to Y_\infty$ such that $\|m_r\| \le M$ for all $r$, and $\phi_r \circ m_r$ is the identity map on $Y_r$.
Question: Suppose $\Gamma \subseteq Y_\infty$ is compact. Does $m_r \circ \phi_r$ converge strongly to the identity operator on $\Gamma$? That is, for all $\epsilon > 0$, does there exist $R > 0$ such that if $r \ge R$, then $$\sup_{y \in \Gamma} \left\| (m_r \circ \phi_r)(y) - y \right\|_{Y_\infty} < \epsilon?$$
