Riehl's basketball-walk enumeration conjecture for increasing unary-binary trees

A basketball walk is a walk with the step set used in the paper, and an increasing unary-binary tree is an increasing tree whose vertices have at most one unary or two binary children, with its associated permutation defined as in the source. For a positive integer $n$, consider basketball walks of length $n$ that start at altitude $0$, end at altitude $1$, and never touch or pass below the $x$-axis.

Riehl's conjecture. The number of basketball walks of length $n$ starting at the origin and ending at altitude $1$ that never touch or pass below the $x$-axis equals the number of increasing unary-binary trees on $n$ vertices with associated permutation avoiding $213$.

The conjecture identifies two enumeration sequences arising from lattice paths and pattern-avoiding increasing trees. It was proved in Bettinelli and Fusy (2016), so the corresponding enumeration is no longer open.