Schur-type arithmetic progression conjecture for primes

Let $k$ be a positive integer and color all primes with $k$ colors. For an integer $l\geq 3$, consider monochromatic primes $p_0,p_1,p_2,\ldots,p_l$.

Schur-type conjecture. For every $l\geq 3$, there exist monochromatic primes $p_0,p_1,p_2,\ldots,p_l$ such that $p_1,\ldots,p_l$ form an arithmetic progression with common difference $p_0-1$.

This conjecture extends the proved Schur-type theorem for primes, which gives monochromatic primes satisfying $p_1+p_2=p_3+1$, and is motivated by the Green–Tao theorem on arbitrarily long arithmetic progressions in the primes. Its resolution is not supplied here.