Fibonacci recurrence-coefficient conjecture for 2-distant and 3-distant noncrossing partitions

Let $NCP_k(n)$ denote the number of $k$-distant noncrossing partitions of $[n]$. For a moment sequence $(\mu_n)_{n\geq 0}$, write its associated monic orthogonal polynomials in the form

$$P_{n+1}(x)=(x-b_n)P_n(x)-\lambda_nP_{n-1}(x).$$

Here $F_n$ denotes the $n$-th Fibonacci number, defined by $F_{n+1}=F_n+F_{n-1}$ and $F_1=F_2=1$.

Fibonacci recurrence-coefficient conjecture. If $\mu_n=NCP_2(n)$, then $b_0=b_1=\lambda_1=1$, and for $n\geq2$,

$$b_n=3-\frac{1}{F_{2n-1}F_{2n-3}},\qquad \lambda_n=1+\frac{1}{(F_{2n-3})^2}.$$

If $\mu_n=NCP_3(n)$, then $b_0=b_1=\lambda_1=1$, $\lambda_2=2$, and, for $n\geq3$, $b_n=3$ and $\lambda_n=2$.

The conjecture gives explicit recurrence coefficients for the orthogonal polynomials associated with the enumerative sequences of 2-distant and 3-distant noncrossing partitions. The preceding computations provide evidence for these formulas, but the source does not state a resolution, so the conjecture remains open.