The panchromatic–bipanchromatic number equality conjecture

Let $H$ be a hypergraph. Write $\chi_p(H)$ for its panchromatic number, $\chi^2_p(H)$ for its bipanchromatic number, and let $\alpha_{\chi_p(H)}(H)$ be the minimum number of unique colors among all panchromatic $\chi_p(H)$-colorings of $H$. The panchromatic–bipanchromatic equality conjecture. Every hypergraph $H$ satisfies

$$\chi^2_p(H)=\chi_p(H)-\left\lceil\frac{\alpha_{\chi_p(H)}(H)}{2}\right\rceil.$$

The equality would give a general relation between panchromatic and bipanchromatic coloring numbers. The source reports computational tests on randomly generated hypergraphs in the stated parameter ranges, but provides no proof or disproof.