Can the difference between consecutive even abundant numbers exceed 6?

I came across an astonishing observation :

An abundant number is a positive integer $n$ with the property $S(n)>n$ , where $S(n)$ is the sum of the divisors of $n$ except $n$ itself.

The difference of consecutive even abundant numbers seems to be at most $6$. Can anyone prove/disprove this statement ?

Difference $6$ is not exceeded upto at least $10^7$.

This can be proven, observing that any multiple of $6$ greater than $6$ itself is abundant: the divisors include at least $1, \frac{n}{2},\frac{n}{3},\frac{n}{6}$, which together sum to $n+1$.