Fractional-power domain conjecture for the operators $A_\alpha$

Let $\Omega$ be the spatial domain, let $\Gamma$ be its boundary, and let $A_\alpha$ denote the operators associated with the strongly damped wave equation for $\alpha\in[0,1]$. For $\theta\in[0,1]$, write $D(A_\alpha^\theta)$ for the domain of the fractional power of $A_\alpha$. Fractional-power domain conjecture. For $\alpha\in(0,1]$, one has

$$D(A_\alpha^\theta)\subset H^1(\Omega)\times H^{\min\{1,2\theta\}}(\Omega)\times H^1(\Gamma)\times H^{\min\{1,2\theta\}}(\Gamma)$$

for all $0\leq\theta\leq1$. For $\alpha=0$, one has

$$D(A_0^\theta)\subset H^1(\Omega)\times H^{\min\{1,2\theta\}}(\Omega)\times H^1(\Gamma)\times H^{\min\{1/2,\theta\}}(\Gamma)$$

for all $0\leq\theta\leq1$. The conjecture would provide a full characterization, or at least precise regularity information, for the fractional-power domains of the operators governing the problem; such information is relevant to extending the well-posedness and attractor results to broader nonlinearities.