Would a monk with the Tavern Brawler feat have improvised weapons count as monk weapons?

Would a monk with the Tavern Brawler feat have improvised weapons count as monk weapons?

It states in the monk’s Martial Arts feature (PHB, p. 78) that:

At 1st level, your practice of martial arts gives you mastery of combat styles that use unarmed strikes and monk weapons, which are short swords and any simple melee weapons that don’t have the two-handed or heavy property.

The Tavern Brawler feat (PHB, p. 170) states that:

You are proficient with improvised weapons and unarmed strikes.

And the rules for improvised weapons (PHB, p. 147-148) state:

In many cases, an improvised weapon is similar to an actual weapon and can be treated as such. For example, a table leg is akin to a club. At the DM’s option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus. An object that bears no resemblance to a weapon deals 1d4 damage.

If the improvised weapon does not resemble a weapon, is simple, and is not heavy or two-handed, could a monk replace the d4 dice? For example if the monk is level 6 and has a unarmed attack of a d6, with a d6 dice instead?