Equating Inv-Quinv formulas for the $q$-Whittaker and modified Hall-Littlewood functions

We explain the equality between the two sets of formulas for $q$-Whittaker functions and modified Hall-Littlewood functions obtained by Haglund, Haiman and Loehr - the Inv formula and Ayyer, Mandelshtam and Martin - the Quinv formula by use of weighted path symmetric functions introduced by Carlsson and Mellit.