Polynomial-time basis-counting conjecture for 2-uniform matroids

Let $M$ be a 2-uniform matroid, given by a representation over the 2-uniform partial field. Basis-counting conjecture. There is a polynomial-time algorithm to count the number of bases of $M$. The paper attributes this conjecture to Hall, Mayhew, and Slilaty and notes that a sufficiently constructive positive resolution of the 2-uniform-to-QU conjecture would settle it.