Axiom of Choice equivalence for colourings of infinite connected graphs

In ZF, let a cardinal number be defined as an equivalence class under equinumerosity. For a graph, its chromatic index is the least cardinal number of colours in a proper edge-colouring; its distinguishing number is the least cardinal number of colours in a vertex-colouring preserved only by the identity automorphism; and its distinguishing chromatic number and distinguishing chromatic index are the corresponding least cardinal numbers subject to both properness and distinguishing requirements. Colouring equivalence conjecture. The following statements are equivalent to the Axiom of Choice in ZF:

1. Every infinite connected graph has a chromatic index.
2. Every infinite connected graph has a distinguishing chromatic number.
3. Every infinite connected graph has a distinguishing chromatic index.
4. Every infinite connected graph has a distinguishing number.
5. Every infinite connected graph has a distinguishing chromatic index.

The question is presented as open in the paper. Its significance is that it seeks to characterize the Axiom of Choice through the existence of cardinal-valued colouring parameters for infinite connected graphs; the duplicated third and fifth clauses should be checked against the source.