OEIS formulas for statistics of image-partition sets

For $n\ge0$, let $\chi(\mathcal S)$ be the number of distinct parts occurring in a set of partitions $\mathcal S$, and let $\tau(\mathcal S)$ count parts with multiplicity. Let $\operatorname{ImP}_k(n)$ and $\operatorname{ImB}_k(n)$ denote the corresponding image sets. OEIS-statistics conjecture. For $n\ge0$,

$$\begin{aligned} \chi(\operatorname{ImP}_2(n))&=A227800(n+1),\\ \chi(\operatorname{ImB}_2(n))&=A126236(n),\\ \chi(\operatorname{ImP}_3(n))&=1+A213213(n),\\ \tau(\operatorname{ImP}_2(n))&=A258472(n). \end{aligned}$$

These identities connect the statistics to named OEIS sequences; the paper supplies no resolution.