Generalized Chern number bounds for matroids

Let $M$ be a simple matroid of rank $d+1$ on a ground set $E$ of size $n$. For nonnegative integers $k_1,\ldots,k_d$ with $k_1+2k_2+\cdots+dk_d=d$, consider the Chern number $\bar{c}_1^{k_1}\cdots \bar{c}_d^{k_d}(M)$ and the corresponding Chern number of the uniform matroid $U_{d+1,n}$. Generalized Chern number bounds conjecture. The Chern numbers of $M$ satisfy

$$|\bar{c}_1^{k_1}\cdots \bar{c}_d^{k_d}(M)| \leq |\bar{c}_1^{k_1}\cdots \bar{c}_d^{k_d}(U_{d+1,n})|.$$

Equality holds if and only if $M=U_{3,n}$. This conjecture proposes that the upper bounds proved for simple matroids of rank $3$ extend to matroids of arbitrary rank; the supplied text does not provide evidence resolving the conjecture.