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 How has this hypothesis been tested? 
 It's part of a model which explains the observed galaxy redshift with very accurate precision and good data fit. 
 How independent is galaxy redshift from whatever phenomena produced the dark energy theory in the first place?

Based on the wiki for Dark Energy, DE was theorized to explain supernova redshift

In which case I am very skeptical that further redshift observations confirm anything positive about the nature of DE as an explanatory factor 
 DE is an integral part of the Standard Cosmology Model as a whole, without it it would fall apart. The LCDM together with the Friedman equations, explain most expansion measurements fairly well.
https://en.m.wikipedia.org/wiki/Lambda-CDM_model