Erdős Problem 850

Reference: erdosproblems.com/850

theorem Erdos850.erdos_850 : True ↔ ∃ (x : ℕ) (y : ℕ), x ≠ y ∧ x.primeFactors = y.primeFactors ∧ (x + 1).primeFactors = (y + 1).primeFactors ∧ (x + 2).primeFactors = (y + 2).primeFactors

Can there exist two distinct integers $x$ and $y$ such that $x,y$ have the same prime factors, $x+1,y+1$ have the same prime factors, and $x+2,y+2$ also have the same prime factors?