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# Metrizability of a topological vector space where every sequence can be made to converge to zero

This is a follow-up to this answer.

If $$E$$ is a (real or complex) topological vector space, we say that a sequence $$\{x_n\}_{n=1}^\infty$$ in $$E$$ can be made to converge to zero if there exists a sequence $$\{\alpha_n\}_{n=1}^\infty$$ of strictly positive real scalars such that $$\lim_{n\to\infty} \alpha_n x_n = 0$$.

In the aforementioned answer, I compared this with two related notions, and I gave a generic example of a space where every sequence can be made to converge to zero: take a metrizable topological vector space and pass to a weaker (i.e. coarser) topology, possibly with a different dual. My question is whether every example is of this form. In other words:

Question. If $$E$$ is a topological vector space where every sequence can be made to converge to zero, is there a finer linear topology on $$E$$ that is metrizable?

Maybe it is false in general, but true in certain special cases? (E.g. if $$E$$ is locally convex/complete/separable, or if $$E$$ the strong dual of a Fréchet space?)