Does the Hexblade Warlock’s Hex Warrior feature apply to a magic weapon that is transformed into your Pact Weapon?

The Hexblade Warlock’s Hex Warrior feature states (emphasis mine):

[…] If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon’s type.

Meanwhile, the Pact of the Blade Warlock feature states:

[…] You can transform one magic weapon into your pact weapon by performing a special ritual while you hold the weapon. You perform the ritual over the course of 1 hour, which can be done during a short rest. You can then dismiss the weapon, shunting it into an extradimensional space, and it appears whenever you create your pact weapon thereafter. […]

Does a magic weapon that you have transformed into your pact weapon benefit from Hex Warrior?
Does it count as a weapon you have conjured using Pact of the Blade?
Does the answer change if you shunt the weapon away and then make it appear afterwards?