Quadratic bound conjecture for products of primitive matrices

Let a set of primitive matrices of dimension $n$ satisfy Assumption 1, namely the nonzero-row, nonzero-column condition stated in the paper. A product of matrices is called positive when all its entries are positive.

Quadratic bound conjecture. There is a constant $K$ such that every such set has a product of length smaller than $Kn^2$ with positive entries.

This conjecture asks whether the general cubic upper bound can be reduced to quadratic order. The preceding discussion notes that the cubic bound is not known to be sharp; the status of the quadratic bound remains open.