 Solution

A reversible Markov chain must satisfy detailed balance. For every $i\ge 1$, the rate from $i$ to $i+1$ is positive, while the reverse rate from $i+1$ to $i$ is zero. No positive measure can satisfy ${\pi }_i{g}_{i,i+1}={\pi }_{i+1}{g}_{i+1,i}$, so the irreducible chain is not reversible.