dnd 5e – Does Mind Sliver’s saving throw penalty apply to Concentration saves from its own damage?

Mind Sliver (Tasha’s Cauldron of Everything p.108) says:

The target must succeed on an Intelligence saving throw or take 1d6 psychic damage and subtract 1d4 from the next saving throw it makes before the end of your next turn.

I am confused about the timing of the “next saving throw”. If I used Mind Sliver on a Concentrating creature and it failed its save, the d6 psychic damage would provoke a Constitution save to maintain Concentration.

Would this save count as its “next saving throw” and be immediately affected by the penalty from the Sliver?

Or would the penalty be applied only after all the effects of the Sliver, including the damage and an unpenalized forced Concentration save, had been resolved?