Gaps between totatives of a primorial: which even numbers occur, and how often? (Erdős #854)

Let $n_k$ be the $k$th primorial, the product $2\cdot 3\cdots p_k$ of the first $k$ primes, and let $1=a_1<a_2<\cdots<a_{\phi(n_k)}=n_k-1$ be the integers in $[1,n_k]$ coprime to $n_k$ (the totatives). Every consecutive gap $a_{i+1}-a_i$ is even. Estimate the smallest even integer that does not occur as any gap $a_{i+1}-a_i$. Moreover, are there $$\gg \max_i (a_{i+1}-a_i)$$ many distinct even integers that do occur as gaps $a_{j+1}-a_j$ (as $k\to\infty$)?

Acceptance. FULLY RESOLVES: a proof determining the asymptotic size of the smallest even integer not occurring as a totative gap of $n_k$, and a proof settling whether the number of distinct realised even gaps is $\gg\max_i(a_{i+1}-a_i)$; OR a proof that Erdős's original guess (every even $t$ up to the maximal gap is realised) fails for infinitely many $k$ (or for all large $k$). ADVANCES: extend the exact computation to primorials $n_k$ beyond the published Lacampagne–Selfridge range — for each new $k$, report with a reproducible program and certificate the maximal gap $g(n_k)$, the smallest even integer not realised as a gap, and the number of distinct even gaps realised, thereby adding data points on whether the count is $\gg g(n_k)$; or prove a non-trivial lower or upper bound on the smallest missing even gap. Deliver the search program plus the per-$k$ tabulated certificates, or the proof.