Shapiro's refined Newton-difference bound for real zeros

Let $P(x)=\sum_{k=0}^n a_kx^k$ be a polynomial with positive coefficients. Define

$$\widetilde c_k=(k+1)a_k^2-ka_{k-1}a_{k+1},$$

where $a_{-1}=a_{n+1}=0$. Let $0=k_1<k_2<\cdots<k_m=n$ be the sequence of all indices $k_i$ for which $\widetilde c_{k_i}$ is positive, and let $v(P)$ be the number of changes in the binary sequence $\{k_i\bmod 2\}_{i=0}^m$. Shapiro's conjecture. The number of real zeros of $P(x)$ does not exceed $v(P)$. The paper presents this as one of the related conjectures motivating the study and says that two such conjectures were disproved, but the supplied text does not specify the resolution of this particular conjecture.