Stephan's third ratio conjecture for the Pascal triangle sequence

Consider Pascal triangle modulo $2$, and let $c(n)$ be the binary number obtained by reading its $n$th row. Define

$$l(n)=\frac{c(2n)-1}{4}.$$

Stephan's third ratio conjecture.

$$\lim_{n\rightarrow\infty}\frac{l(8n+4)}{l(8n+3)}=\frac{257}{85}.$$

The conjecture belongs to R. Stephan's series concerning the sequence $\{l(n)\}_{n\geq 0}$. The paper's abstract says that these conjectures are proved, but the supplied candidate does not include the relevant proof or resolution evidence.