Duchi–Guerrini–Rinaldi–Schaeffer's refined fighting fish–ternary tree conjecture

A fighting fish is considered together with its size, fin length, number of tails, and numbers of ascending and descending strips. A left ternary tree is considered together with its number of nodes, core size, number of right branches, and numbers of non-root nodes with even abscissa and nodes with odd abscissa. The fin of a fighting fish is the path starting from the left point of its head, following its border counterclockwise, and ending at the right point of the first tail reached. The core of a ternary tree is the largest subtree including the root vertex and consisting only of left and middle edges.

Duchi–Guerrini–Rinaldi–Schaeffer's conjecture. The number of fighting fish with size $n$, fin length $k$, having $h$ tails, $i$ ascending strips and $j$ descending strips is equal to the number of left ternary trees with $n$ nodes, core size $k$, having $h$ right branches, $i+1$ non-root nodes with even abscissa and $j$ nodes with odd abscissa.

This conjecture refines the established enumerative correspondence between fighting fish and ternary trees by matching several statistics simultaneously. The source states that it remains unresolved.