Hyperbolicity theorem conjecture for local girth correspondence colouring

Let local girth correspondence colouring refer to correspondence colouring with list sizes varying according to local girth. A hyperbolicity theorem is a theorem of the type used in the paper's main hyperbolicity result to control the relevant graph families and precolouring extensions.

Local girth correspondence-colouring hyperbolicity conjecture. There exists a theorem analogous to the paper's main hyperbolicity theorem for local girth correspondence colouring.

The source presents this as the correspondence-colouring analogue of the proposed local girth list-colouring hyperbolicity theorem, and indicates that it is not known. It remains open.