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Let f (x) be a monic polynomial with integer coefficients and 0 ≤ r1 ≤ ··· ≤ rn<p its roots modulo a prime p. We generalize a conjecture on the distribution of roots ri with additional congruence relations ri ≡ Ri mod L from the case that f has no non-trivial linear relation among roots to the case that f has a non-trivial linear relation.