Forsgård–Shapiro coefficient-difference bound for real zeros

Let $f(z)=\sum_{k=0}^n a_kz^k$ be a polynomial with positive coefficients. Set $a_{-1}=a_{n+1}=0$ and

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

Let $0=k_1<k_2<\dots<k_m=n$ be the indices for which $\tilde c_{k_i}>0$, and let $v(f)$ be the number of changes in the sequence of parities $\{k_i\bmod 2\}_{i=0}^m$.

Forsgård–Shapiro coefficient-difference conjecture. The number of real zeros of $f$ does not exceed $v(f)$.

The source presents this as an open conjecture based on experimental motivation.