Andrews' smallest-parts moment inequality

Let $N(m,n)$ and $M(m,n)$ denote the rank and crank counting functions for partitions, and for even $k$ define the rank and crank moments

$$N_k(n)=\sum_m m^kN(m,n),\qquad M_k(n)=\sum_m m^kM(m,n).$$

Andrews' smallest-parts moment inequality. For $k$ even, $k\geq2$, and $n\geq1$,

$$M_k(n)>N_k(n).$$

This inequality would strengthen the known relation $\operatorname{spt}(n)=\tfrac12(M_2(n)-N_2(n))$ and give positivity of the smallest-parts function through a broader hierarchy of rank and crank moment inequalities. The supplied text gives no resolution.