Euler–Kronecker constant analogue of Kummer's conjecture

Let $q$ be a prime, and let $\kappa(q)$ denote the Euler–Kronecker constant quantity defined in the paper for the maximal real cyclotomic subfield of $\mathbb Q(\zeta_q)$. The Euler–Kronecker analogue of Kummer's conjecture.

$$\kappa(q)=o(1)\qquad\text{as }q\to\infty.$$

This is proposed as an analogue of Kummer's prediction for the relative class number. The paper states conditional results showing that this claim fails under suitable Elliott–Halberstam and Hardy–Littlewood hypotheses, but does not establish an unconditional resolution.