The infinite Toeplitz trace conjecture for return probabilities

Consider the Markov chain on $\mathbb{N}$ in which every vertex $i$ has an edge of weight $r$ to $i-1$ and edges of weight $c_l$ to $i+l$ for $0\leq l\leq k$, with

$$r+\sum_{i=0}^{k}c_i\leq 1.$$

Let $p_l$ be the probability of starting at $i$ and returning to $i$ in exactly $l$ steps. Infinite Toeplitz Trace Conjecture. For some constant $\alpha$, if

$$p_l\leq \frac{\alpha}{k}\quad\text{for }1\leq l\leq k+1,$$

then

$$p_l\leq \frac{\alpha}{k}\quad\text{for all }l.$$

The paper explicitly describes this infinite version as open and as a relaxation of the finite Toeplitz trace conjecture.