Symmetric-difference formula conjecture for the EMD numerator

For positive integers $n$, let $N_n(t)$ be the numerator polynomial in

$$H'_n(t)=\frac{N_n(t)}{(1-t)^{2n}},$$

and define

$$S(m)=\sum_{(X,Y)\in 2^{[m]}\times 2^{[m]}}|X\mathbin{\triangle}Y|=m\,2^{2m-1},$$

where $2^{[m]}$ is the power set of $[m]$. Symmetric-difference formula conjecture. For all positive integers $n$, we have

$$N_n(1)=S(n-1).$$

The identity was subsequently proved in the cited Ding preprint, although the authors note that a bijective proof remains especially desirable.