Positivity and uniqueness conjecture for the recurrence coefficients

Let $(a_n,b_n)$ satisfy the recurrence relations referred to as --, with $a_0=0$ and $n\in\mathbb{N}$. A solution is called positive when $a_{n+1}>0$ and $b_n>0$ for every $n\in\mathbb{N}$.

Positivity and uniqueness conjecture. There is a unique positive solution, and it corresponds to the initial condition

$$b_0=\frac{\Gamma(2/3)}{\Gamma(1/3)}.$$

This conjecture concerns the large-$n$ behavior of recurrence coefficients associated with multiple orthogonal polynomials for an exponential cubic weight. The supplied text gives numerical evidence for positivity but provides no resolution of the uniqueness claim.