Dokos–Dwyer–Johnson–Sagan–Selsor coefficient-parity conjecture for 321-avoiding permutations

For each integer $k\geq 0$, let $M_{2^k-1}(321;q)$ denote the major-index generating polynomial associated with 321-avoiding permutations of length $2^k-1$, and write $\langle q^i\rangle$ for the coefficient of $q^i$. Dokos–Dwyer–Johnson–Sagan–Selsor's coefficient-parity conjecture. For all $k\geq 0$,

$$\langle q^i\rangle M_{2^k-1}(321;q)= \begin{cases} 1 & \text{if } i=0,\\ \text{an even number} & \text{if } i\geq 1. \end{cases}$$

This conjecture concerns the parity of the coefficients in the major-index generating polynomial for 321-avoiding permutations at lengths one less than powers of two. The source attributes it to Dokos, Dwyer, Johnson, Sagan and Selsor as their Conjecture 3.6; no resolution is given here.