Invariance of reflecting ordinals under iterated Sacks, Miller, and Laver forcing

Let $L$ be the constructible universe, and let $\theta$ range over the reflecting ordinals $\sigma^1_n$ and $\pi^1_n$. Consider forcing over $L$ with countable-support iterations of Sacks forcing, Miller forcing, or Laver forcing. Forcing-invariance conjecture. The values of $\sigma^1_n$ and $\pi^1_n$ remain unchanged after forcing over $L$ with countable-support iterations of Sacks forcing. The same holds for Miller and Laver forcing. This conjecture extends the corresponding preservation results for a single forcing notion to countable-support iterations; whether all three classes of iterations preserve these reflecting ordinals remains open.