Modular Kraus entropic conjecture for Parseval frames

Let $\mathcal{E}$ be a Hilbert $C^*$-module over a commutative unital $C^*$-algebra $\mathcal{A}$. Let $\{\tau_j\}_{j=1}^\infty$ and $\{\omega_k\}_{k=1}^\infty$ be two Parseval frames for $\mathcal{E}$. For $x\in\mathcal{E}_\tau\cap\mathcal{E}_\omega$, let $S_\tau(x)$ and $S_\omega(x)$ denote the entropies associated with the two frames. Modular Kraus entropic conjecture. One has

$$S_\tau(x)+S_\omega(x)\geq -2\log\left(\sup_{j,k\in\mathbb{N}}\|\langle\tau_j,\omega_k\rangle\|\right),\qquad \forall x\in\mathcal{E}_\tau\cap\mathcal{E}_\omega.$$

This would extend the Maassen–Uffink entropic uncertainty principle and the Ricaud–Torrésani version from finite-dimensional Hilbert spaces and Parseval frames to Parseval frames in Hilbert $C^*$-modules. The source motivates the conjecture by noting that the usual proofs rely on Riesz–Thorin interpolation, which is not known there for abstract Hilbert $C^*$-modules.