Interpolation conjecture for two-indexed Schatten quasi-norms

Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces, and let $\mathcal{S}_p(\mathcal{H})$ denote the Schatten class of index $p$. For $\theta,r\in[0,1]$ and $1\leq q_1,p_1\leq\infty$, define $q,p$ by

$$\frac{1}{q}=\frac{1-\theta}{r}+\frac{\theta}{q_1},\qquad \frac{1}{p}=\frac{1-\theta}{r}+\frac{\theta}{p_1}.$$

Interpolation conjecture. The quasi-Banach spaces defined in the paper are complex interpolation spaces, completely isometrically, in the sense that

$$\operatorname{\mathcal{S}}_q[\mathcal{H}_1,\operatorname{\mathcal{S}}_p(\mathcal{H}_2)] = \left[\operatorname{\mathcal{S}}_{r}(\mathcal{H}_1\otimes\mathcal{H}_2),\operatorname{\mathcal{S}}_{q_1}[\mathcal{H}_1,\operatorname{\mathcal{S}}_{p_1}(\mathcal{H}_2)]\right]_\theta.$$

This conjecture extends the known interpolation-scale property of Pisier norms and the interpolation-scale property of Schatten quasi-norms to the two-indexed Schatten quasi-norms introduced in the paper. Its resolution would establish a complex interpolation framework for these quasi-Banach spaces beyond the currently stated results.