Gun–Murty–Rath transcendence conjecture for shifted reciprocal-power series

Let $\alpha$ be a non-integral rational number and let $k>1$ be a natural number. Define the shifted reciprocal-power series

$$\sum_{n=-\infty}^{+\infty}\frac{1}{(n+\alpha)^k}.$$

Gun–Murty–Rath conjecture. The number

$$\sum_{n=-\infty}^{+\infty}\frac{1}{(n+\alpha)^k}$$

is transcendental. The paper presents this as a conjecture by Gun, Murty and Rath and states that it is false when $k$ is odd and $\alpha$ is a half-integer; it further claims that these are the only counterexamples, so the conjecture as stated is refuted.