Rational generating functions for grid classes with forest graphs

Let $M$ be a finite matrix defining a grid class $\operatorname{Grid}(M)$, and let $G(M)$ be the bipartite graph whose vertices represent the rows and columns of $M$, with an edge whenever the corresponding entry of $M$ is nonzero. A generating function is rational if it is a quotient of polynomials. Forest rationality conjecture. If $G(M)$ is a forest, then $\operatorname{Grid}(M)$ and all its subclasses have rational generating functions. The source describes this as a speculative extension of known results: permutation-matrix grid classes have eventually polynomial enumeration, while star-graph grid classes have rational generating functions. Its resolution is not given in the source.