The greedy Tamari interval–constellation bijection conjecture

Let $m$ be a positive integer. Consider greedy $m$-Tamari intervals whose maximal element has $n_i$ ascents of length $i$ for each $i\ge 1$, including a first ascent of length $\ell$. Consider $(m+1)$-constellations with $n_i$ white faces of degree $(m+1)i$ for each $i\ge 1$, including a white root face of degree $(m+1)\ell$. The greedy Tamari interval–constellation conjecture. The number of such greedy $m$-Tamari intervals equals the number of such $(m+1)$-constellations. This conjecture seeks a bijective explanation of the enumerative equality proved earlier in the paper; finding such a bijection remains open.