An equivalent condition for second countable locally convex spaces

Question

Let $(X,\tau)$ be a locally convex topological vector space.

Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming from $\mathcal{E}$ and $\tau$ are the same. Can we conclude that $X$ is second countable ?!

Generally, if $X$ is just a topological space, the answer will be negative. See A criterion for second countability