The answer
Yes. Reconstruction can also require the multisig configuration: cosigner extended public keys, script type and derivation information.
What to do next
Export and securely store a complete wallet descriptor or recovery configuration alongside key backups.
Trezor hardware wallets
Potential cold-storage product; not a substitute for safe backup procedures. Availability, supported assets and local eligibility must be checked before signing up or buying.
Visit Trezor hardware wallets ↗Official provider link · Compare suitability and availability before choosing.