Item text formatting for name/lore is not upgraded correctly from 1.20.2 if quotes are used on true/false

If an item has a name or lore with a formatting option such as bold or italics set to true/false with quotes, it isn't properly upgraded from 1.20.2 to newer versions.

Upgrading from 1.20.2 to any version from 23w40a through 24w07a retains the name/lore with formatting but it no longer renders.
Upgrading from 1.20.2 to any version from 24w09a through 1.20.6 strips any name/lore with quotes present on a true/false value.
Upgrading from 1.20.2 to 24w18a+ strips any formatting with quotes present on true/false, but retains the name/lore.

While the use of quotes here is technically incorrect, this is something that the DFU should probably catch and fix to avoid damaging custom items on maps and servers.

"italic":"false" does not upgrade correctly, while "italic":false does upgrade correctly.

Steps to reproduce:
On 1.20.2, run the following commands:
    /give @s cookie{display:{Name:'

{"text":"Item with quotes on true/false","color":"yellow","italic":"false"}

'}}
    /give @s cookie{display:{Name:'

{"text":"Item without quotes on true/false","color":"yellow","italic":false}

'}}

You should have 2 items that both have correctly rendering names:

If upgraded from 1.20.2 to 1.20.3/4, or any version from 23w40a to 24w07a, the name of item with the quotes around true/false no longer renders, while the item without quotes renders correctly:

If upgraded from 1.20.2 to 1.20.5/6, or any 1.20.5 snapshot starting with 24w09a, the name of the item with the quotes is permanently lost, while the item without quotes upgrades correctly:

If upgraded from 1.20.2 to 1.21, or any version 24w18a+, the name of the item with the quotes is retained, but it is now italicized when it shouldn't be, while the item without quotes upgrades correctly: