Conjecture on weighted frame representations of invertible unconditionally convergent multipliers

Let ${\cal H}$ be a Hilbert space, let $\Phi=(\varphi_n)$ and $\Psi=(\psi_n)$ be sequences in ${\cal H}$, and let $m=(m_n)$ be a complex sequence. The multiplier $M_{m,\Phi,\Psi}$ is given by

$$M_{m,\Phi,\Psi}f=\sum_n m_n\langle f,\psi_n\rangle\varphi_n.$$

It is unconditionally convergent if this series converges unconditionally for every $f\in{\cal H}$. A sequence in ${\cal H}$ is a frame for ${\cal H}$ if it satisfies the frame inequalities, and a multiplier is invertible when it is an invertible bounded operator on ${\cal H}$. Weighted frame representation conjecture. Let $M_{m,\Phi,\Psi}$ be an unconditionally convergent multiplier which is invertible on ${\cal H}$. Then there exist $(\alpha_n)\subset\mathbb C$ and $(\beta_n)\subset\mathbb C$ such that $(\alpha_n\varphi_n)$ and $(\beta_n\psi_n)$ are frames for ${\cal H}$ and

$$\alpha_n\overline{\beta_n}=m_n$$

for all $n$. This asks whether every invertible unconditionally convergent multiplier admits a representation using weighted versions of its two defining sequences that are both frames; the context states that this question was posed as a conjecture, while its general status is not otherwise resolved here.