Currie–Saari recurrence conjecture for unbordered factors of the Thue–Morse sequence

Let $\mathbf{t}=t_0t_1t_2\cdots$ be the Thue–Morse sequence, defined by $t_0=0$, $t_{2n}=t_n$, and $t_{2n+1}=1-t_n$ for $n\geq 0$. Let $f(n)$ denote the number of unbordered factors of length $n$ in $\mathbf{t}$. Currie–Saari's conjecture. The initial values are

$$f(0)=1,\qquad f(1)=2,\qquad f(2)=2,$$

and, for $n\geq 0$, $f$ satisfies

$$\begin{aligned} f(4n+1)&=f(2n+1),\\ f(8n+2)&=f(2n+1)-8f(4n)+f(4n+3)+4f(8n),\\ f(8n+3)&=2f(2n)-f(2n+1)+5f(4n)+f(4n+2)-3f(8n),\\ f(8n+4)&=-4f(4n)+2f(4n+2)+2f(8n),\\ f(8n+6)&=2f(2n)-f(2n+1)+f(4n)+f(4n+2)+f(4n+3)-f(8n),\\ f(16n)&=-2f(4n)+3f(8n),\\ f(16n+7)&=-2f(2n)+f(2n+1)-5f(4n)+f(4n+2)+3f(8n),\\ f(16n+8)&=-8f(4n)+4f(4n+2)+4f(8n),\\ f(16n+15)&=-8f(4n)+2f(4n+3)+4f(8n)+f(8n+7). \end{aligned}$$

The conjecture seeks an explicit description of the number of unbordered factors of each length; the preceding characterization determines which lengths occur but not their multiplicities. Its status is unresolved in the supplied source context.