What a stronger result would require

No such new environmental bound is claimed here. Fixing $L$ excludes only that declared register class; excluding classical environments of arbitrary size needs a bound uniform over the admitted sizes.

Once laboratory instruments are fixed, a process witness may be linear in a process representation. That observation does not automatically turn an unrestricted joint optimization over processes, instruments, and classical-memory decompositions into a convex problem. Any relaxation and its upper-bound status must be justified.