The Craven–Csordas–Smith conjecture on real zeros of a polynomial Wronskian

Let $P$ be a real polynomial, and suppose that $P$ has $2s$ nonreal zeros. Define its Wronskian with its derivative by

$$W(P,P')=PP”-(P')^2.$$

Craven–Csordas–Smith conjecture. The polynomial $W(P,P')$ has at most $2s$ real zeros.

This conjecture concerns the relation between the nonreal zeros of a real polynomial and the real critical behavior of its logarithmic derivative. The source presents it as an intriguing conjecture motivating the study of real polynomial pencils; its resolution is not specified here.