Forsgård–Shapiro tropical bound for real zeros

Let $f(z)=\sum_{k=0}^n a_kz^k$ be a polynomial with positive coefficients, and define

$$f_{\operatorname{trop}}(x)=\max_k\left(\operatorname{Log}(a_k)+kx+\operatorname{Log}{n\choose k}\right).$$

Forsgård–Shapiro tropical conjecture. The number of real zeros of $f$ does not exceed the number of corners of the continuous piecewise-linear function $f_{\operatorname{trop}}$ on $\mathbb{R}$.

The assertion is proposed as a weighted tropical analogue of a real-root bound; the source gives no resolution.