Definitely not fully independent. This paper has some data on the correlation between false-negative % and viral load:
in all groups, CareStart sensitivity followed Ct value distribution, with 96.3% sensitivity observed in all participants with Ct ≤25 and 79.6% in those with Ct ≤30. Although false-negative CareStart results were largely confined to those perhaps least likely to transmit SARS-CoV-2, the sensitivity of the CareStart test by Ct threshold cutoff was lower than observed in our recent study of the Abbott BinaxNOW in the same testing site [6] (99.3% with Ct ≤25, 95.8% with ≤30, and 81.2% with ≤35).
Definitely not fully independent. This paper has some data on the correlation between false-negative % and viral load:
(Higher Ct values indicate lower viral loads.)