Continuous Kraus conjecture for continuous Parseval frames

Let $(\Omega,\mu)$ and $(\Delta,\nu)$ be measure spaces, let $\mathcal{H}$ be a Hilbert space, and let $\{\tau_\alpha\}_{\alpha\in\Omega}$ and $\{\omega_\beta\}_{\beta\in\Delta}$ be $1$-bounded continuous Parseval frames for $\mathcal{H}$. For $h\in\mathcal{H}_\tau\cap\mathcal{H}_\omega$, write $S_\tau(h)$ and $S_\omega(h)$ for the corresponding frame entropies. Continuous Kraus conjecture. Then

$$S_\tau(h)+S_\omega(h)\geq -2\log\left(\sup_{\alpha\in\Omega,\,\beta\in\Delta}|\langle\tau_\alpha,\omega_\beta\rangle|\right)\geq 0,$$

for all $h\in\mathcal{H}_\tau\cap\mathcal{H}_\omega$. This conjecture proposes a continuous-frame analogue of the Kraus entropic uncertainty principle; its status is not resolved in the supplied source context.