  We establish an equidistribution between the pairs (inv,maj) and (quinv,maj)
on any row-equivalency class $[\tau]$ where $\tau$ is a filling of a given
Young diagram. In particular if $\tau$ is a filling of a rectangular diagram,
the triples (inv,quinv,maj) and (quinv,inv,maj) have the same distribution over
$[\tau]$. Our main result affirms a conjecture proposed by Ayyer, Mandelshtam
and Martin (2021), thus presenting the equivalence between two refined formulas
for the modified Macdonald polynomials.
