The quasi-quarter-turn FPL link-pattern distribution conjecture

Let $A_{\textsc{QT}}(4n+2;w)$ denote the number of qQTFPLs of size $4n+2$ with the induced half-turn-invariant link pattern $w$ of length $2n+1$, and let $A_{\textsc{HT}}(2n+1;w)$ denote the number of half-turn-invariant FPLs of size $2n+1$ with link pattern $w$. Let $A(n)$ be the total number of ordinary FPLs of size $n$. The qQTFPL link-pattern conjecture. For any $n\geq 0$ and any half-turn-invariant link pattern $w$ of length $2n+1$,

$$A_{\textsc{QT}}(4n+2;w)=A_{\textsc{HT}}(2n+1;w)A(n+1)A(n).$$

This conjecture gives a link-pattern refinement of the qQTFPL enumeration product formula, relating qQTFPLs of size $4n+2$ to half-turn-invariant FPLs of size $2n+1$. The source gives no resolution evidence.