Convergence of left continuous inverse

Let $f$ be a non-decreasing function. The left continuous inverse of $f$ is given by $f^{\leftarrow}(x) = \inf \{s \colon f(s) \geq x \}$

In the book Extreme value theory: an introduction, there's the following lemma.

Suppose $f_n$ is a sequence of non-decreasing functions and $g$ is a non-decreasing function. Suppose that for each $x$ in open interval $(a,b)$ that is a continuity point of $g$,

$$\lim_{n \to \infty} f_n(x) = g(x)$$.

Let $f_n^{\leftarrow}$ and $g^{\leftarrow}$ be the left-continuous inverse of $f_n$ and $g$. Then for each $x$ in the interval $(g(a), g(b))$ that is a continuity point of $g^{\leftarrow}$, we have

$$\lim_{n \to \infty} f_n^{\leftarrow}(x) = g^{\leftarrow}(x)$$.

Is the converse of this lemma also true? i.e. does the convergence in left-continuous inverse on all continuous points gives the convergence in the original function in all continuous points?

The proof in the case of CDF can be found here: convergence in distribution and convergence of quantile

However, I believe the proof in the cited textbooks and the other answers use that the CDF is a bounded function since it uses inversion sampling. For example, what if the range of $f$ was some infinite interval?

Thank you.

de Haan, Laurens; Ferreira, Ana, Extreme value theory. An introduction., Springer Series in Operations Research and Financial Engineering. New York, NY: Springer (ISBN 0-387-23946-4/hbk). xvi, 417 p. (2006). ZBL1101.62002.