Bijection conjecture for fixed points of the involution on beta(0,1)-trees

Let $g$ be the involution on beta$(0,1)$-trees. For each $n$, consider beta$(0,1)$-trees on $n$ nodes fixed under $g$, and beta$(0,1)$-trees on $\lfloor n/2\rfloor+1$ nodes with one distinguished excessive node.

Bijection conjecture. There is a bijection between beta$(0,1)$-trees on $n$ nodes fixed under $g$ and beta$(0,1)$-trees on $\lfloor n/2\rfloor+1$ nodes with one distinguished excessive node.

The conjecture reformulates the fixed-point enumeration conjecture: the preceding proposition gives $a(n)$ for the latter class when the number of nodes is $n+1$. No proof of this bijection is supplied.