Will the equity from my house be taken into account?

I’ve been paying regular full CSA payments for 10 years. I want to move house now to be with my new partner. There is 130k equity in the house. I understand equity in your primary residence is treated as a capital disregard.

I may need to rent for a few months until the house I’m buying can complete. I did read on the gov work and pensions site that capital taken from sale of primary residence and ear marked for purchase of a new primary residence will be treated as a capital disregard for up to one year. Can anyone confirm if the CSA are bound by this rule please?