In this paper, Adamczak defines, for $m>0$ and $\sigma\geq 0$, the class of probability distributions $\mathcal{M}(m,\sigma)$ over $\mathbb{R}$ as those $\mu$ satisfying the tail conditions $$\nu^+([x,\infty)) \leq \mu([x,\infty)),\quad \nu^-((-\infty,-x]) \leq \mu((-\infty,-x])$$ for every $x\geq m$, where $\nu^+,\nu^-$ are the measures with densities $g^+(x) = x\mu([x,\infty)$ and $g^-(x) = -x\mu((-\infty,-x])$, respectively. (See Definition 4, or the followup work by Adamczak and Strzelecki which generalizes the results of the first paper)
The paper also proves (Lemma 2) that the condition $\mathcal{M}(m,\sigma)$ implies $C(m,\sigma)$-subgaussianity for some constant $C(m,\sigma)$. Some more examples are given, but I am not fully seeing how this class $\mathcal{M}(m,\sigma)$ is related to subgaussianity:
How much stricter is the condition $\mathcal{M}(m,\sigma)$ than subgaussianity? Are there "natural" examples of subgaussian distributions which are not in $\mathcal{M}(m,\sigma)$ for any $m,\sigma$?
Moreover, in view of one of the main results of the paper, Theorem 3 (the log-Sobolev inequality for convex functions holds for measures in $\mathcal{M}(m,\sigma)$):
Is it known whether the subgaussian concentration inequality in Theorem 3 fails for some random variables which are sugaussian but not in any $\mathcal{M}(m,\sigma)$?