Does Shillelagh affect a quarterstaff’s versatile damage?

I was looking at Shillelagh, which includes the following effects.

The wood of a club or quarterstaff you are holding is imbued with nature’s power.

the weapon’s damage die becomes a d8. The weapon also becomes magical, if it isn’t already. (PHB, pg. 275)

Now, the club is fine, it just goes from a 1d4 to a 1d8. However, a quarterstaff does 1d6 and has versatility, letting you wield it two-handed for 1d8.

Since Shillelagh increases the quarterstaff’s base damage to a 1d8, would it be feasible to say that the versatile damage increases to a 1d10?

I’m basing this off the fact that all weapons that do 1d8 damage and have the versatile property (battleaxe, longsword, warhammer) all have their versatile damage as a 1d10 (PHB, pg. 149).