Measure minimization conjecture for continuous frames

Let $(\Omega, \mu)$ be a measure space, let $\mathcal{H}$ be a Hilbert space, and let $\\{\tau_\alpha\\}_{\alpha\in\Omega}$ be a continuous frame for $\mathcal{H}$ with synthesis operator $\theta_\tau^*$. For $h\in\mathcal{H}$, suppose that

$$h=\theta_\tau^*f$$

for some $f\in\mathcal{L}^2(\Omega,\mu)$ satisfying

$$\mu(\operatorname{supp}(f))<\frac{1}{2}\left(1+\frac{1}{\displaystyle\sup_{\alpha,\beta\in\Omega,\alpha\neq\beta}|\langle\tau_\alpha,\tau_\beta\rangle|}\right).$$

Measure minimization conjecture. Then $f$ is the unique solution of

$$\operatorname{minimize}\\{\mu(\operatorname{supp}(g)):g\in\mathcal{L}^2(\Omega,\mu)\\}$$

subject to $h=\theta_\tau^*g$.

This conjecture extends the Donoho–Elad sparsity theorem from finite frames and discrete coefficient support to continuous frames and measure of coefficient support. Its status is not established by the supplied text.