Linear upper-bound conjecture for the SIA-index

Let $n$ be a positive integer and let $\mathcal{S}$ be an SIA set of $n \times n$ stochastic matrices. Its SIA-index is the length of the shortest SIA product formed from matrices in $\mathcal{S}$. Linear upper-bound conjecture. The SIA-index of a set of $n \times n$ stochastic matrices is bounded by $2n$. This would substantially improve the known cubic upper bound and would be consistent with the linear lower-bound evidence described in the paper. The conjecture remains open in the supplied source.