Continuity characterization by preservation of connected sets for arbitrary polynomials

Let $E$ be an infinite-dimensional normed space, and let $P\in\mathcal P_{n,a}(E)$ be an arbitrary polynomial. Condition (2) is the condition that $P(C)$ is connected for every connected subset $C$ of $E$.

Connected-set continuity conjecture. Condition (2) characterizes continuity: $P$ is continuous if and only if $P(C)$ is connected for every connected subset $C$ of $E$.

The paper establishes this characterization for homogeneous polynomials of degrees $1$ and $2$, while the corresponding assertion for arbitrary polynomials is presented as a belief and remains open here.