Enumeration of minimal factors of the form $xxx^R$

Let $F_m$ be the Fibonacci numbers, let $\bf r$ be the Rote-Fibonacci word, and let $(p_i)_F$, $(q_i)_F$, and $(s_i)_F$ denote the specified Fibonacci representations of the starting positions $p_i$, $q_i$, and $s_i$. Consider finite words of the form $xxx^R$ having no proper factor of the form $www^R$.

Minimal-$xxx^R$ enumeration conjecture. For $n=F_{3k+1}$ there are $4$ such words of length $n$. For $n=F_{3k+1}\pm F_{3k-2}$ there are $2$ such words. Otherwise there are none. For $k\geq 3$, the words are exactly the factors of $\bf r$ beginning at the positions whose Fibonacci representations are

$$(p_1)_F=1000(010)^{k-3}001,\quad (p_2)_F=10(010)^{k-2}001,$$ $$(p_3)_F=1001000(010)^{k-3}001,\quad (p_4)_F=1010(010)^{k-2}001,$$

for length $F_{3k+1}$;

$$(q_1)_F=10(010)^{k-3}001,\quad (q_2)_F=10000(010)^{k-3}001,$$

for length $F_{3k+1}-F_{3k-2}$; and

$$(s_1)_F=10(010)^{k-3}001,\quad (s_2)_F=1000(01)^{k-2}001,$$

for length $F_{3k+1}+F_{3k-2}$.

This is a detailed conjectural enumeration of the shortest, or factor-minimal, occurrences of $xxx^R$ in the Rote-Fibonacci word. It follows an explicit open problem in a section collecting claims the authors had not yet proved, so its status is open.