File descriptor from flock put in a variable is not found when releasing it with exec

Question

I'm trying to get a lock on a file with bash. I've found some docs, including this nice answer:

Linux flock, how to "just" lock a file?

It works, BUT I cannot release the file descriptor using a variable.

This WORKS

LOCK_FILE="lock.lock"
# Acquire lock - lock_fd is always 10 in my machine
exec {lock_fd}>$LOCK_FILE || exit 1
flock "$lock_fd" || exit 1
# Release lock - OK
exec 10>&-

See the last line to release the lock: exec 10>&-. If I put the number it works

BUT, this DOES NOT work:

LOCK_FILE="lock.lock"
# Acquire lock
exec {lock_fd}>$LOCK_FILE || exit 1
flock "$lock_fd" || exit 1
# Release lock - not found
exec $lock_fd>&-

I use the variable exec $lock_fd>&-, as shown in the answer above (and in some other documentation I've found around)

./lock.sh: line 6: exec: 10: not found

Somebody could tell what is wrong? Thank you.