Ostrovsky and Paes Leme's matroid-based valuation conjecture

Let $v:2^{\mathcal G}\to\mathbb R$ be a valuation, and define the merging operation (or convolution) of valuations $v_1,v_2$ by

$$v^*(S)=\max_{T\subseteq S} v_1(T)+v_2(S\setminus T)\quad\forall S\subseteq\mathcal G.$$

An $M^\natural$-concave valuation is a valuation in the discrete-concavity class denoted by $M^\natural$-concavity, and a weighted matroid rank function is a valuation of that form. Ostrovsky and Paes Leme's matroid-based valuation conjecture. Every $M^\natural$-concave valuation arises by the repeated application of endowment and merging operations starting from weighted matroid rank functions. The source identifies this as a stronger conjecture previously posed by Ostrovsky and Paes Leme and states that it is still open; merging preserves $M^\natural$-concavity.