Does the disadvantage on saves from Shadow Sorcerer’s Hound of Ill Omen feature persist beyond the first turn?

Shadow Sorcerer’s Hound of Ill Omen feature’s text reads:

…Additionally, while the hound is within 5 feet of the target, the target has disadvantage on saving throws against any spell you cast. The hound disappears if it is reduced to 0 hit points, if its target is reduced to 0 hit points, or after 5 minutes.

I’m curious if against any spell you cast refers to the action of casting the spell, or the condition of having been a spell that you cast in any tense.

The difference being that a spell like Hold Person would only get disadvantage on the first saving throw with the first interpretation, but on all following saving throws with the second interpretation.

The only comparable feature that I know of is Heightened Spell, a Sorcerer metamagic, that reads:

When you cast a spell that forces a creature to make a saving throw to resist its effects, you can spend 3 sorcery points to give one target of the spell disadvantage on its first saving throw made against the spell.

This feature is clear in its intent, that only the initial saving throw against any spell is affected and given disadvantage.

The lack of clarification on Hound of Ill Omen makes me wary of which way to rule.