The Hawaiian conjecture on chords of the garden of a logarithmic derivative

Let $P$ be a real polynomial of degree $n$, and let $\mathcal{G}_{P}$ be the garden of the rational function $P'/P$. A chord of $\mathcal{G}_{P}$ is a chord in this garden; the source also notes that all zeros of $P$ lie on $\mathcal{G}_{P}$.

Hawaiian conjecture. Each chord of $\mathcal{G}_{P}$ contains at least one nonreal zero of $P$.

The conjecture is formulated as a topological strengthening of the zero bound for the Wronskian $W(P,P')$: the endpoints of the chords are the real zeros of that Wronskian. The source states that this conjecture would immediately imply the Hawaii conjecture, but gives no resolution.