Do any maximally inflected curves exist?      
Ramification is simple if $ \alpha\in\{(m,0,\ldots), (1,\ldots,1,0\ldots)\}$
Theorem. [Real Schubert Calculus]   (S.-)
    When all ramification is simple, there exists a choice of real ramification points
    for which all linear series are real.

Conjecture. (Shapiro-Shapiro)
    For any choice of real ramification, all linear series are real.

Theorem. (Gabrielov-Eremenko)
    (1) Conjecture is true when $ k=2$ or $ d=k+2$.
    (2) In those cases, when $ d$ is even, purely imaginary ramification gives no real linear series.