Rotemberg in a DCP/LCP layout

[HelloDynare]

@HelloDynare

Furthermore, do I need change \mathcal{F}_t to:
\mathcal{F}_{t} = \left[(1-\gamma)^{\frac{1}{\theta}}{\tilde{Q}_{HHt}}^{\frac{\theta-1}{\theta}}+\gamma^{\frac{1}{\theta}}{\tilde{Q}_{FHt}}^{\frac{\theta-1}{\theta}}\right]^{\frac{\theta}{\theta-1}}
in which, because of the Rotemberg friction / deadweight loss,
\tilde{Q}_{HHt}= \displaystyle\frac{Q_{HHt}}{1+\frac{\xi_P}{2}\left[\frac{P_{HHt}(\omega)}{ P_{HH,t-1}(\omega)}-\bar{\Pi}_{HH}\right]^2}
and
\tilde{Q}_{FHt}= \displaystyle\frac{Q_{FHt}}{1+\frac{\xi_P}{2}\left[\frac{P_{FHt}(\omega)}{ P_{FH,t-1}(\omega)}-\bar{\Pi}_{FH}\right]^2}
??

I find one recent paper goes along this line as well. But the treatment is more complicated than my previous post.