A tropical analog of Descartes' rule of signs

We prove that for any degree d, there exist (families of) finite sequences a_0, a_1,..., a_d of positive numbers such that, for any real polynomial P of degree d, the number of its real roots is less than or equal to the number of the so-called essential tropical roots of the polynomial obtained from P by multiplication of its coefficients by a_0, a_1,... a_d respectively. In particular, for any real univariate polynomial P of degree d with non-vanishing constant term, we conjecture that one can take a_k = e^{-k^2}, k = 0, ... , d. The latter claim can be thought of as a tropical generalization of Descartes's rule of signs. We settle this conjecture up to degree 4 as well as a weaker statement for arbitrary real polynomials. Additionally we describe an application of the latter conjecture to the classical Karlin problem on zero-diminishing sequences.