Let $\varphi$ denote Euler's totient function. It is easy to see that all those numbers $$\varphi(n^2)=n\varphi(n)\ \ (n=1,2,3,\ldots)$$ are pairwise distinct.

I have the following surprising conjecture.

**Conjecture.** Any positive rational number $r$ has the form $\varphi(m^2)/\varphi(n^2)$ with $m$ and $n$ positive integers.

I have verified this for $r\in\{a/b:\ a,b=1,\ldots,50\}$. My computation shows that \begin{align}&\left\{\frac{\varphi(m^2)}{\varphi(n^2)}:\ m,n=1,\ldots,15000\right\}\\\supseteq&\left\{\frac ab:\ 1\le a,b\le 50\ \&\ \{a,b\}\not=\{19,47\},\{37,47\}\right\}.\end{align} In addition, I have found that $$\frac{\varphi(12765^2)}{\varphi(18612^2)}=\frac{80879040}{102738240} =\frac{37}{47}$$ and $$\frac{\varphi(39330^2)}{\varphi(55836^2)}=\frac{373792320}{924644160} =\frac{19}{47}.$$

I have no good explanation for the conjecture, but I'm confident that it should be true.

**QUESTION:** Is the above conjecture true? Are there any supporting heuristic arguments?

Your further check of the conjecture is also welcome!