Monoid conjecture for bracelet matrices

Let $\mathcal{L}_d$ be the set of bracelet matrices of dimension $d$. A monoid is a set closed under matrix multiplication and containing an identity element $\mathbb{I}_d$. Monoid conjecture for bracelet matrices. The set $\mathcal{L}_d$ forms a monoid: it is closed under matrix multiplication and contains the identity matrix $\mathbb{I}_d$. This conjecture is supported by numerical investigations; the paper proves the weaker result that $\mathcal{L}_d$ is closed under multiplication by factorisable matrices and contains every factorisable bistochastic matrix.