nt.number theory - Inequality with Euler's totient

Using the first 100,000 values of $varphi(n)$ it seems that the following is true.

Let $mathcal A$ be a finite subset of $mathbb{N}$, $forall nin mathbb{N} setminus mathcal{A}$, $displaystyle frac{1}{varphi(2n)} - frac{1}{varphi(2n+1)} geqslant frac{1}{2nln (2n)} $.

Is this true? Is there a stronger lower bound?

P.S.: I looked at Handbook of Number Theory I by Mitrinović and Sándor which has a lot of info about $varphi (n)$ but it doesn't appear there.