Bergelson–McCutcheon IP-limit positivity conjecture for VIP-systems

Let $\mathcal H$ be a Hilbert space and let $(V_\xi)_{\xi\in{\mathbb F_2^\omega}}$ be a VIP-system of unitary operators on $\mathcal H$. Suppose that for every $f\in\mathcal H$ there is a vector $Qf\in\mathcal H$ such that

$$\mathop{\operatorname{IP-lim}}_{\xi\in{\mathbb F_2^\omega}}V_\xi f=Qf$$

weakly.

Bergelson–McCutcheon IP-limit positivity conjecture. For every $f\in\mathcal H$,

$$\operatorname{Re}\langle f,Qf\rangle\ge 0,$$

or, equivalently,

$$\mathop{\operatorname{IP^*-lim}}_{\xi\in{\mathbb F_2^\omega}}\operatorname{Re}\langle f,V_\xi f\rangle\ge 0.$$

The paper presents this as an adaptation of Conjecture 2 of Bergelson and McCutcheon; no resolution is given in the supplied text.