The quarter-turn FPL link-pattern distribution conjecture

Let $A_{\textsc{QT}}(4n;w)$ denote the number of quarter-turn-invariant FPLs of size $4n$ with link pattern $w$, where $w$ is a bilateral Dyck word of length $2n$, and let $A_{\textsc{HT}}(2n;w)$ denote the corresponding number for half-turn-invariant FPLs of size $2n$. Let $A(n)$ be the total number of ordinary FPLs of size $n$. The quarter-turn link-pattern conjecture. For any $n\geq 0$ and bilateral Dyck word $w$ of length $2n$,

$$A_{\textsc{QT}}(4n;w)=A_{\textsc{HT}}(2n;w)A(n)^2.$$

Equivalently, even-sized quarter-turn-invariant FPLs are conjectured to have exactly the link-pattern distribution of half-turn-invariant FPLs with half their size, up to the factor $A(n)^2$. The conjecture was checked by exhaustive enumeration through size $20$; the source says that further exhaustive generation is unreasonable.