Highly Connected Counterexamples to the Multiplicative Merino–Welsh Inequality
For every integer \(k\ge 3\), we construct \(k\)-connected matroids representable over every infinite field, with girth \(k\), arbitrarily large cogirth, and Merino–Welsh quotient \[ \Phi(M)=\frac{T_M(1,1)^2}{T_M(2,0)T_M(0,2)} \] tending to infinity. In particular, this disproves **Conjecture 7.2 of Csikvári (2026)**, which asserts the multiplicative Merino–Welsh inequality for simple and cosimple matroids, and therefore also the stronger **Conjecture 7.3** therein, as observed by its author. Our examples are rank truncations of direct sums of \(k\)-element circuits. We give a 3-connected…
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