Shapiro–Shapiro conjecture for maximally inflected rational curves

Let $d>r$ be integers and let $calpha^1,\ldots,\alpha^n$ be ramification data for degree $d$ rational curves in $\mathbb{P}^r$. Let $s_1,\ldots,s_n$ be distinct points of $\mathbb{R}\mathbb{P}^1$. Shapiro–Shapiro conjecture. Every degree-$d$ rational curve $\varphi:\mathbb{P}^1\rightarrow\mathbb{P}^r$ with ramification $\alpha^i$ at $s_i$ for each $i=1,\ldots,n$ is real. The question of whether real ramification data always produce real curves is unknown in general; this conjecture would give a strong affirmative answer and is a formulation of the Shapiro–Shapiro conjecture in real Schubert calculus.