Endpoint complex interpolation conjecture for Sobolev spaces on quantum tori

Let $\alpha_0, \alpha_1\in{\mathbb R}$ and $1<p<\infty$. For the Sobolev spaces $H_{\infty}^{\alpha_0}({\mathbb T}^d_\theta)$, $H_{1}^{\alpha_1}({\mathbb T}^d_\theta)$, and $H_{p}^{\alpha}({\mathbb T}^d_\theta)$ on the quantum torus, set

$$\alpha=\left(1-\frac1p\right)\alpha_0+\frac{\alpha_1}{p}.$$

Endpoint complex interpolation conjecture.

$$\big(H_{\infty}^{\alpha_0}({\mathbb T}^d_\theta),\, H_{1}^{\alpha_1}({\mathbb T}^d_\theta)\big)_{\frac1p}=H_{p}^{\alpha}({\mathbb T}^d_\theta).$$

This extends the preceding complex interpolation result to the endpoint couple involving $H_{\infty}$ and $H_1$. In the commutative case, the corresponding K-functional is known, but determining the complex interpolation spaces of this couple is a longstanding open problem; the quantum-torus assertion is therefore open.