Logarithmic Sobolev inequality for the noncommutative two-torus

Let $A_{\theta}^{\infty}$ be the smooth noncommutative two-torus with canonical trace $\tau$, generated by unitaries $U,V$, and let $a=\sum_{(m,n)\in\mathbb{Z}^{2}}a_{m,n}U^{m}V^{n}\in A_{\theta}^{\infty}$ satisfy $a>0$. Write $\|a\|_{2}^{2}=\tau(a^{2})$. The logarithmic Sobolev conjecture. One has

$$\tau(a^{2}\log a)\leq \sum_{(m,n)\in\mathbb{Z}^{2}}(|m|+|n|)|a_{m,n}|^{2}+\|a\|_{2}^{2}\log\|a\|_{2}.$$

Equivalently,

$$\tau(a^{2}\log a)\leq \sum_{(m,n)\in\mathbb{Z}^{2}}(|m|+|n|)|a_{m,n}|^{2}+\tau(a^{2})\log(\tau(a))^{1/2}.$$

The paper presents this as the main conjectured logarithmic Sobolev inequality on the noncommutative $2$-torus and proves it for certain elements, while the general assertion remains unresolved in the supplied text.