Pantone's 123-avoiding derangement proportion conjecture

Let $C_n$ be the $n$th Catalan number, and let $d_n$ be the number of derangements among the $123$-avoiding permutations of length $n$. The source considers an asymptotic form $d_n\sim \delta C_n$ for a limiting proportion $\delta$. Pantone's conjecture. The limiting proportion of derangements among $123$-avoiding permutations is equal to $9/16$. This is a numerical conjecture based on the first $200$ enumeration terms and differential approximants.