  The matroid-based valuation conjecture of Ostrovsky and Paes Leme states that
all gross substitutes valuations on $n$ items can be produced from merging and
endowments of weighted ranks of matroids defined on at most $m(n)$ items. We
show that if $m(n) = n$, then this statement holds for $n \leq 3$ and fails for
all $n \geq 4$. In particular, the set of gross substitutes valuations on $n
\geq 4$ items is strictly larger than the set of matroid based valuations
defined on the ground set $[n]$. Our proof uses matroid theory and discrete
convex analysis to explicitly construct a large family of counter-examples. It
indicates that merging and endowment by themselves are poor operations to
generate gross substitutes valuations. We also connect the general MBV
conjecture and related questions to long-standing open problems in matroid
theory, and conclude with open questions at the intersection of this field and
economics.
