Growth of $\tau_\perp(n)$, the count of coprime consecutive divisors of $n$ (Erdős #1100)

For $n\geq 2$ write the divisors of $n$ in increasing order as $1=d_1<d_2<\cdots<d_{\tau(n)}=n$, and let $\tau_\perp(n)$ count the indices $i$ (with $1\leq i<\tau(n)$) for which consecutive divisors are coprime, i.e. $\gcd(d_i,d_{i+1})=1$. Let $\omega(n)$ denote the number of distinct prime factors of $n$. Three questions are asked. (i) Is $\tau_\perp(n)/\omega(n)\to\infty$ for almost all $n$? (ii) Is it true that $$\tau_\perp(n)<\exp\big((\log n)^{o(1)}\big)$$ for all $n$? (iii) Restricting to squarefree $n$, determine the growth of $$g(k)=\max_{\substack{\omega(n)=k\\ n\ \text{squarefree}}}\tau_\perp(n).$$

Acceptance. FULLY RESOLVES: complete proofs (machine-checkable in Lean/Coq preferred, otherwise full written proofs) answering any one of the three questions in full — for (i), whether $\tau_\perp(n)/\omega(n)\to\infty$ for almost all $n$; for (ii), whether $\tau_\perp(n)<\exp((\log n)^{o(1)})$ holds for all $n$; for (iii), the true exponential growth rate, i.e. the value of $\lim_k g(k)^{1/k}\in[\sqrt2,\,2-c]$ (or matching upper and lower bounds pinning it). ADVANCES: strictly narrow the Erdős–Simonovits window $(2^{1/2}+o(1))^k<g(k)<(2-c)^k$ by raising the lower base above $\sqrt2$ or lowering the upper base below the best value implied in the background, with proof; OR compute $g(k)$ exactly for a range of small $k$ not previously tabulated, with a reproducible enumeration certificate, refining the conjectured base; OR prove a partial result toward (i) or (ii), e.g. $\tau_\perp(n)/\omega(n)\to\infty$ on a density-1 subset under a stated hypothesis. Deliver the proof file, the improved-bound proof, or the enumeration code plus the table of exact $g(k)$ values.