The CAPP conjecture for weighted matroid measures

Let a weighted matroid measure be obtained by assigning positive external-field weights to the independent sets of a matroid. For a measure on $2k$ variables, define the antipodal pairs property (APP) by

$$\alpha_k(\mu)\geq\alpha_{k-1}(\mu),$$

where $\alpha_i(\mu)={n\choose i}^{-1}\sum_{|\eta|=i}\mu(\eta)\mu(\underline{1}-\eta)$; define CAPP by requiring APP after conditioning on any $n-2k$ variables. Weighted-matroid CAPP conjecture. Weighted matroid measures have CAPP. This is presented as a possible strengthening of Mason's conjecture, with no resolution supplied.