Does uniform convergence of $F_n*G_t$ entail uniform convergence of $F_n\,$?

Question

Consider a sequence of functions $F_n\in C_c(\mathbb R^d)$ for $n\in\mathbb N\,$ and a function $F\in C_c(\mathbb R^d)\,$, all sharing the same compact support $K$. Let $(G_t)_{t>0}$ be an approximation of the identity, precisely $$ G_t(x) :=\, \frac{1}{(2\pi t)^{d/2}}\;e^{-|x|^2/(2 t) }\,.$$ Suppose that for any fixed $t>0$, $$ \|(F_n- F)* G_t\|_\infty \,\to\, 0 \quad\textrm{as }n\to\infty \;.$$ Can we conclude that $$ \|(F_n- F)\|_\infty \,\to\, 0 \quad\textrm{as }n\to\infty \;?$$

Answer 1

No, it doesn't follow. A hint for constructing a very simple counterexample: note that $$||(F_n-F)*G_t||_\infty\le||F_n-F||_2||G_t||_2.$$