The finite-dimensionality conjecture for type spaces of finitely forcible graphons

Let $T(W)$ denote the type space associated with a graphon $W$. Finite-dimensionality conjecture. If $W$ is finitely forcible, then $T(W)$ is finite dimensional. The source intentionally leaves unspecified which notion of dimension is intended; any corresponding finite-dimensionality result would be of interest. In the examples considered, $T(W)$ is either $0$-dimensional or $1$-dimensional, while the linear hull of $T(W)$ is generally infinite dimensional unless $T(W)$ is finite.