An infinite set of totient values whose smallest preimages grow superlinearly? (Erdős #51)

Let $\varphi$ be Euler's totient function. Is there an infinite set $A\subseteq\mathbb{N}$ such that every $a\in A$ is a totient value (there exists an integer $n$ with $\varphi(n)=a$), and yet, writing $n_a$ for the smallest integer with $\varphi(n_a)=a$, one has $$\frac{n_a}{a}\to\infty\quad\text{as } a\to\infty\ (a\in A)?$$ In words: can an infinite family of totient values have smallest preimages that are eventually arbitrarily large relative to the value itself?

Acceptance. FULLY RESOLVES: EITHER exhibit such an infinite set $A$ — an explicit description or construction of infinitely many totient values $a$, a proof that $\varphi(n)=a$ is solvable for each, and a proof that the smallest-preimage ratio $n_a/a\to\infty$ along $A$ — OR a complete proof that no such infinite $A$ exists (for every infinite set of totient values, $n_a/a$ does not tend to infinity). ADVANCES: prove existence under a stated hypothesis; OR construct an infinite set of totient values along which $n_a/a\to\infty$ on a subsequence, or with a quantified growth $n_a/a\gg g(a)$ for an explicit unbounded $g$ on an infinite set, with proof; OR deliver a reproducible computation exhibiting totient values with record-large $n_a/a$, extending the known data with the inversion code and certified ratios. Deliver the construction plus proofs, or the non-existence proof, or the record-search code plus certified ratios.