Lando's continued-fraction conjecture for the universal $sl_2$ weight system

Let $D_n(x)$ denote the polynomial associated with the $n$-chord diagram, and define

$$F(x,t)=\sum_{n\geq 0}D_n(x)t^n.$$

For $k\geq 0$, set $b_k(x)=x-k(k+1)$, and for $k\geq 1$ set $\lambda_k(x)=-k^2x+\binom{k}{2}\binom{k+1}{2}$. Lando's continued-fraction conjecture. The generating function $F(x,t)$ has the continued-fraction expansion

$$F(x,t)=\dfrac{1}{1-b_0(x)t-\dfrac{\lambda_1(x)t^2}{1-b_1(x)t-\dfrac{\lambda_2(x)t^2}{\ddots}}}.$$

This is presented as a broader conjecture of which the proved congruence above is a particular case; the supplied text gives no resolution of the full continued-fraction assertion.