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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?)

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