Does the bard’s Jack of All Trades feature make them proficient with thieves’ tools?

PHB describes the Jack of All Trades feature as

you can add half your proficiency bonus, rounded down, to any ability check you make that doesn’t already include your proficiency bonus

It also says that

trying to open a lock requires proficiency with thieves’ tools

Assuming that Jack of All Trades does not give you proficiency in any skill that you are not proficient in, does it mean a bard actually can NOT pick a lock?