Safe deposit boxes are regulated by state statutes, state court decisions, and the terms of the lease; although there are some best practices, there are no universal answers.
It seems unlikely that your first two resources would address this issue, but no one could guarantee that without looking at them. However, look carefully at the actual terms of your lease. All of those that require employee signatures on termination that I have seen are so the employee can attest to the fact that the box was inspected and found to be empty.
If that's the way yours is worded you would be assuming unnecessary risks not to require your employees to sign them.
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In this world you must be oh so smart or oh so pleasant. Well, for years I was smart. I recommend pleasant.