Tiling conjecture for minimal diskbusting lists of words

Let $F_n$ be a free group, and let a list of cyclically reduced words in $F_n$ be minimal and diskbusting, meaning that it cannot be decomposed into a proper sublist with the same relevant polygonality property and that no nontrivial free product decomposition $F_n=A\ast B$ has every word conjugate into one of the factors. A list is polygonal when it admits the polygonal disk-and-side-pairing structure used to construct a surface in the double $D(U)$. Tiling conjecture. Every minimal and diskbusting list of cyclically reduced words in $F_n$ is polygonal when $n>1$. This conjecture would extend the polygonality criterion beyond the cases currently established and, together with the polygonality theorem, would produce hyperbolic surface subgroups in the corresponding doubles.