Euler's totient function $\varphi$ is multiplicative, and it plays important roles in number theory.

QUESTION: Is it true that for each integer $m>6$ we have $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$ for some positive integer $n$?

For every $m=7,\ldots,10^4$, I have found the smallest positive integer $n$ such that $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$ (cf. http://oeis.org/A248007). For example, for $m=10$ the least positive integer $n$ with $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$ is $14$; in fact, $$\varphi(10)\varphi(14)=4\times 6\equiv0\pmod{10+14}.$$

I formulated the above question in 2014 and conjectured that the answer should be positive. I have ever mentioned this question to some number theorists, but there is no substantial progress on the question.

I don't think that the question is very difficult. Any ideas towards its solution?