Twisted Connes–Popa rigidity conjecture for ICC property (T) groups

Let $G$ be an ICC property (T) group, let $H$ be an arbitrary countable ICC group, and let $d:G\times G\to\mathbb{T}$ and $c:H\times H\to\mathbb{T}$ be $2$-cocycles. For $t>0$, suppose that

$$\Psi:\mathcal{L}_d(G)^t\to\mathcal{L}_c(H)$$

is a $*$-isomorphism. Twisted Connes–Popa rigidity conjecture. Then $t=1$, there is a group isomorphism $\delta:G\to H$ such that $d$ is cohomologous to $c\circ\delta$, and there exist a map $\xi:G\to\mathbb{T}$, a multiplicative character $\eta:G\to\mathbb{T}$, and a unitary $w\in\mathcal{L}_d(H)$ satisfying

$$d(g,h)=\xi_g\xi_hc(\delta(g),\delta(h))\overline{\xi_{gh}}$$

and

$$\Psi(u_g)=\eta_g\xi_gwv_{\delta(g)}w^*,\qquad\text{for all }g\in G.$$

This is proposed as an extension of Popa's strengthening of Connes' rigidity conjecture to twisted group factors. The source gives preceding results supporting the formulation but does not state that it is resolved.