The equivalence of abundance for graphs and coloured graphs

Let $F$ be a graph, let $H$ be a graph, and let an $F$-colouring of $H$ be a colouring $\sigma$ for which $(H,\sigma)$ is an $F$-coloured graph. The graph $H$ is $F$-abundant when it has the corresponding abundance property, and $(H,\sigma)$ is $F$-abundant when the coloured graph has that property.

Abundance equivalence conjecture. A graph $H$ is $F$-abundant if and only if there is an $F$-colouring $\sigma$ of $H$ such that $(H, \sigma)$ is $F$-abundant.

The conjecture asks whether proving abundance for some colouring is not only sufficient but also necessary for the underlying graph to be $F$-abundant. The surrounding discussion identifies this as a central open question in the classification of abundance.