For a topological space $X$ and a point $x\in X$, consider the following definitions:

- (Gerlits and Nagy): $X$ is strictly Fréchet at $x\in X$ if for any sequence $(A_n)_{n\in\omega}$ such that $x\in\bigcap_{n\in\omega}\overline{A_n}$ there exists a sequence $(x_n)_{n\in\omega}\in\prod_{n\in\omega}A_n$ such that $x_n\to x$.
- (Siwiec): $X$ is strongly Fréchet at $x\in X$ if for any
*decreasing*sequence $(A_n)_{n\in\omega}$ such that $x\in \bigcap_{n\in\omega}\overline{A_n}$ there exists a sequence $(x_n)_{n\in\omega}\in\prod_{n\in\omega}A_n$ such that $x_n\to x$.

Clearly, strictly Fréchet spaces are strongly Fréchet. Since these conditions are equivalent for spaces of the form $C_p(X)$, I was wondering if the converse holds in general. More precisely,

If $X$ is strongly Fréchet at $x$, then $X$ is strictly Fréchet at $x$?