Solvability of a system related to the subsets of {1,2,3}, II

In the first post: Solvability of a system related to the subsets of {1,2,3}, we have shown (here) that an allowed labeling $f$ of $B_3$ can have a negative Euler totient $\varphi(f)$ [equals for example to $-1/4 + 3/100$ for $f$ below].

We have also proved that for any allowed labeling $f$ of $B_3$ then $\varphi(f) > -1$.

So the natural question is now the following:
Question: What's the infimum of $\{ \varphi(f) \mid f$ an allowed labeling $f$ of $B_3 \}$? Is it $-1/4$? What's the proof?

According to Mathematica, this infimum is exactly $-1/4$:
gives a sequence $f_n$ such that $\varphi(f_n)$ converges to the infimum: take $x=9/4$ and $\epsilon_n = 1/n$.