Lexicographically minimal maximal sparse paving matroids have exponential extension complexity

For each positive integer $n$, let $\mathsf{S}_n$ be the sparse paving matroid on $[n]$ of rank $\lfloor n/2\rfloor$ with the maximal possible number of circuit-hyperplanes and lexicographically minimal set of bases. The extension-complexity conjecture. For every positive integer $n$, its base polytope satisfies

$$\operatorname{xc}(\mathscr{P}(\mathsf{S}_n)) \in \Omega\left(\frac{2^{n/2}}{n^{5/4}\sqrt{\log(2n)}}\right).$$

Rothvoss proved the existence of matroids with this asymptotic lower bound, but his argument is non-constructive; an explicit exponential family remains open. The conjecture proposes that this particular explicitly selected family has the desired extension complexity, with the construction of $\mathsf{S}_n$ itself also noted as an open problem.