How integrity is compromised if forward secrecy is not enabled in TLS communication?

If TLS communication uses ciphers that does not support forward secrecy(FS) (like RSA key exchange ciphers), confidentiality of the past communication is compromised if the private key is compromised. But will the integrity also gets compromised in this scenario? I got this doubt after seeing the CVSS scoring in this link. Vector: CVSS:3.0/AV:N/AC:H/PR:N/UI:N/S:U/C:H/I:L/A:N – The severity of integrity compromise is given as Low. Could anyone explain how integrity is getting affected in the absence of FS?