Space of cadlag functions - Nonexistence of a TVS Polish topology?

Question

Consider the space $D := D([0,1], \mathbb{R})$ of real-valued cadlag functions on $[0,1]$. On $D$ one can invent several (more or less) well-known topologies:

1. the uniform topology $U$: $(D, U)$ is a Banach space which is not separable (thus not Polish)
2. the Skorokhod $J_1$-topology: $(D, J_1)$ is Polish but not a topological vector space (TVS) since addition is not continuous everywhere
3. the weaker Skorokhod $M_1$-topology: $(D, M_1)$ is also Polish but again not a TVS (addition is not continuous everywhere but the set of discontinuities is smaller than for $J_1$)
4. the Skorokhod $J_2$ topology (weaker than $J_1$) and $M_2$ topology (weaker than $M_1$) : $D$ is Lusin (I think not Polish) and not a TVS again
5. the Jakubowski $S$-topology: $(D,S)$ is Lusin, not metrizable (thus not Polish), addition is sequentially continuous, but not continuous everywhere and thus $(D,S)$ ist not a TVS.

This leads me to the following:

Conjecture: On $D$ one can not define a topology such that $D$ is both Polish and a TVS (or even more a separable Banach space).

Does anyone know some attempts to show this or is it an open problem?