Finite-difference moment identities for elementary symmetric partition images

Let $\mathcal P(n)$ be the set of partitions of $n$, let $\mathcal P[m,n]=\biguplus_{i=m}^n\mathcal P(i)$, and let $\operatorname{ImP}_k[m,n]=\biguplus_{i=m}^n\operatorname{ImP}_k(i)$. Let $m_j$ denote the $j$th moment statistic and let $\Delta$ denote the finite-difference operator. Finite-difference moment conjecture. For $n\ge0$ and $k\ge2$,

$$\begin{aligned} \Delta^{k-1}m_4(\operatorname{ImP}_k[n,n+1])&=m_2(\mathcal P(n))+m_4(\mathcal P[n,n+1]),\\ \Delta^{k-1}m_6(\operatorname{ImP}_k(n))&=m_6(\mathcal P(n))+m_2(\mathcal P(n-2))-m_3(\mathcal P(n-2)),\\ \Delta^{k-1}m_6(\operatorname{ImP}_k[n,n+1])&=m_3(\mathcal P(n))+m_6(\mathcal P[n,n+1]),\\ \Delta^{k-1}m_9(\operatorname{ImP}_k[n,n+2])&=m_3(\mathcal P(n))+m_9(\mathcal P[n,n+2]),\\ \Delta^{k-1}m_{10}(\operatorname{ImP}_k[n,n+1])&=m_5(\mathcal P(n))+m_{10}(\mathcal P[n,n+1]),\\ \Delta^{k-1}m_p(\operatorname{ImP}_k(n))&=m_p(\mathcal P(n)), \end{aligned}$$

where the last equality holds for $p=1$ or $p$ prime. These identities are proposed as a direction for future work, and the paper gives no resolution.