The bounded-support-point conjecture for normalized Loewner mappings

Let $S^0(\mathcal D_n)$ be the compact class of normalized mappings generated by Loewner chains, and call $f\in S^0(\mathcal D_n)$ a support point if a nonconstant real part of a continuous linear functional attains its maximum at $f$. Bounded-support-point conjecture. If $f\in S^0(\mathcal D_n)$ is bounded, then $f$ is not a support point of $S^0(\mathcal D_n)$. In one complex variable the analogous support points are unbounded; the assertion is presented as open in higher dimensions.