Multi-sector NK model with different growth rate

Dear users,

I am trying to working on a two-sector NK model with different growth rate, The model has a investment good which is a CES function comined with two sector’s goods:
in (Carvalho et al. 2021) they assume the steady-state levels of productivities are the same across sectors,so detrending the Investment good can be done.My question is :if two sector’s productivities exhibt different growth rate, will the model have a steady-state?

I thank you in advance for your help.

My guess is that the answer is no. Depending on the elasticity of substitution, one sector would become increasingly bigger or smaller than the other one.

Thank you for your reply!