  In the context of the dispersion relation $c = \lambda \nu$ and considering
an expanding universe where the observed wavelength today is redshifted from
the emitted wavelength by $\lambda_{0} = \lambda_{\text{emit}} (1+z)$, to keep
$c$ constant, it must be that $\nu_{0} = \nu_{\text{emit}} /(1+z)$. However,
although the theory for wavelength in the RW metric includes the cosmological
redshift, the same is not simply deduced for frequency (the inverse of time).
Instead, cosmological time dilation $T_{0} = T_{\text{emit}} (1+z)$ is an
additional assumption made to uphold the hypothesis of constant speed of light
rather than a relation directly derived from the RW metric. Therefore,
verifying cosmological time dilation observationally is crucial. The most
recent data employing supernovae for this purpose was released recently by the
Dark Energy Survey. Results from the i-band specifically support variations in
the speed of light within 1-$\sigma$. We used these observations to investigate
variations in various physical quantities, including $c$ and $G$, using the
minimally extended varying speed of light model. The speed of light was $0.4$\%
to $2.2$\% slower, and Newton's constant may have decreased by $1.7$\% to
$8.4$\% compared to their current values at redshift $2$. These findings,
consistent with previous studies, hint at resolving tensions between different
$\Lambda$CDM cosmological backgrounds but are not yet conclusive evidence of a
varying speed of light, as the full-band data aligns with standard model
cosmology.
