Shapiro's weighted tropical bound for real zeros

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

$$\operatorname{ftrop}(P)(t)=\max_k\left(\log(a_k)+kt+\log\binom{n}{k}\right).$$

A corner is a point where this piecewise-linear function is not differentiable. Shapiro's conjecture. The number of real zeros of $P(x)$ does not exceed the number of corners of $\operatorname{ftrop}(P)(t)$ for $t\in\mathbb{R}$. The paper states that this conjecture was among the motivations for the work and that two related conjectures were subsequently disproved, but it does not identify which conjecture this concerns or provide a resolution here.