Mao's identities for Beck's partition statistics modulo 8

Let $NT(r,m,n)$ denote the total number of parts in partitions of $n$ whose rank is congruent to $r$ modulo $m$, and let $M_{\omega}(r,m,n)$ denote the total number of ones in partitions of $n$ whose crank is congruent to $r$ modulo $m$. For $n\geq 0$, Mao's conjectured identities.

$$NT(2,8,4n)-NT(6,8,4n)=M_{\omega}(1,4,4n)-M_{\omega}(3,4,4n),$$ $$NT(6,8,4n+2)-NT(2,8,4n+2)=M_{\omega}(1,4,4n+2)-M_{\omega}(3,4,4n+2).$$

These are the two remaining identities from five identities conjectured by Mao; the present paper proves them, completing the conjecture.