This depends on each individual case and each Insurance company.
I have never seen the General Damages increased in order to prevent a rejection of an Assessment.
I have seen cases where items of Special Damage were paid after the Assessment was sent to both sides. For example Physiotherapy bills were paid directly by the Insurance company in addition to what was submitted in the claim as the vouchers were not available before the Assessment was made. A phone call to the Insurance company suggesting the Injured party might accept the Assessment if the Insurance paid these “extra” vouchers is usually worthwhile.