Interpolation stability conjecture for the scale ${}_{\Sigma}L_p$

Let ${}_{\Sigma}L_p$ denote the family of spaces associated with the surface $\Sigma$, and let $[\,\ ]_{\theta}$ and $(\,\ )_{\theta,r}$ denote the complex and real interpolation methods, respectively. Interpolation stability conjecture. The scale ${}_{\Sigma}L_p$ is stable under interpolation:

$$[{}_{\Sigma}L_p,{}_{\Sigma}L_q]_{\theta}={}_{\Sigma}L_r,\qquad ({}_{\Sigma}L_p,{}_{\Sigma}L_q)_{\theta,r}={}_{\Sigma}L_r,$$

where

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

The conjecture seeks interpolation theorems for the spaces ${}_{\Sigma}L_p$; the source suggests interpolation techniques may be useful but gives no resolution.