Let $X$ be a vector space. A vector (not necessarily Hausdorff) topology on $X$ will be called convex sequential if every convex sequentially closed subset of $X$ is closed.

Is there some description of class of convex sequential topologies?

There is a description of C-sequential topologies (i.e. every convex sequentially open subset is open; for example, http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002307847&physid=PHYS_0280), but I can't find anything about convex sequential topologies.

In particular,

1) Is there a convex sequential, but not sequential topology?

2) Let $X$ bу a topological vector space such that all linear functionals on $X$ are continious. Can the topology on $X$ be convex sequential, but not sequential?

EDIT: I found an answer to questions 1 and 2. Let $X$ be an infinite-countable dinensional vector space, $X^\#$ its algebraic dual space. Then the weak topology relative to the duality $X \leftrightarrow X^\#$ is convex sequential, but not sequential. However, the main question is not solved.