For a locally convex Hausdorff spaces $E$, consider the canonical map $$\overline{\psi}:E^\prime \hat{\otimes}_\pi E \longrightarrow L(E_\sigma)$$ that maps the projective tensor product to the space of weakly continuous maps on $E$ (note that if $E$ is Banach spaces, then the latter is just the space $L(E)$ of bounded linear operators).

The question whether this map is injective is well-known to be connected to the approximation property. For example:

• If $E$ is a Banach space, then $E$ has the approximation property if and only if $\overline{\psi}$ is injective.

• If $E$ has is a complete locally convex space with a fundamental system of absolutely convex neighborhoods $U$ of zero such that every Banach space $E_U$ has the approximation property, then $\overline{\psi}$ is injective.

These statements can be found in Köthe's book "Topologcial Vector Spaces II".

Question: If $E$ is complete and has the approximation property, is it false/true/not known whether $\overline{\psi}$ is injective in general?