Is every Lindelöf quasi-metrizable space separable?

Question

A Hausdorff space $(X,\tau)$ is quasi-metrizable iff there is a function $g: \omega \times X \to \tau$ such that

(i) $\{g(n,x): n\in \omega\}$ is a local base at $x$;

(ii) $y\in g(n+1, x) \implies g(n+1, y) \subset g(n,x)$.

As we know, every Lindelöf metrizable space is separable. But,

Is every Lindelöf quasi-metrizable space separable?

Thanks for your help.