The Merino-Welsh conjecture (Merino and Welsh 1999) asserts that every connected graph 
without graph loops or graph
bridges satisfies
 |
(1)
|
Here 
is the Tutte polynomial. The three quantities
count graph orientations having no directed
cycles, graph orientations
in which every edge lies on a directed cycle,
and spanning trees, respectively.
The multiplicative Merino-Welsh conjecture (Conde and Merino 2009) is the stronger inequality
 |
(2)
|
It remains open for graphs. Its direct extension to matroids without loops or coloops was refuted by Beke et al. (2024). A matroid loop belongs to no matroid basis, while a coloop belongs to every matroid basis.
Csikvári (2026) conjectured a sharp universal replacement for matroids. Let 
be the largest real polynomial root of
 |
(3)
|
Liu (2026) reported a proof that every finite matroid 
of this type satisfies
 |
(4)
|
Known counterexamples for every 
make 
the least nonnegative universal parameter. More precisely,
if 
has 
elements and 
connected components, then for 
,
 |
(5)
|
Examples in Liu (2026) show that the exponential rate is sharp throughout this interval.
Liu reports that GPT-6 Astra and Claude Opus 5 assisted with mathematical exploration, proof development, exact computation, and preparation of the manuscript and code. As of Sep. 27, 2026, the result had not undergone external peer review, and the rational replacement table used by the proof had not been independently recomputed.
