Identify stochastic matrices from their product

Question

Let $K, T$ positive integers. For $t\in\{1,\dots,T\}$, let $Q(t)$ a $K\times K$ stochastic matrix. We assume that for all $t$, $Q(t)$ is invertible and irreducible.

Let $$M(1) = Q(1)Q(2)\cdots Q(T),$$ $$M(2) = Q(2)Q(3)\cdots Q(T)Q(1)$$ $$\vdots$$ $$M(T) = Q(T)Q(1)\cdots Q(T-1).$$

Are $Q(1),\dots,Q(T)$ uniquely determined by $M(1),\dots,M(T)$ ?

Answer 1

The short answer is no, at least when $T=2$. E.g. we have $AB=BA=XY=YX=M$ when $$ \ A=\frac14\pmatrix{3&1\\ 1&3}, \ B=\frac16\pmatrix{5&1\\ 1&5}, \ X=\frac15\pmatrix{4&1\\ 1&4}, \ Y=\frac19\pmatrix{7&2\\ 2&7}, \ M=\frac13\pmatrix{2&1\\ 1&2}. $$