I am working on a related project involving Grimm's conjecture.  The hope is to show that every interval of consecutive composite numbers below $10^{12}$ contains an injective divisor map, see https://mathoverflow.net/q/248146 for more detail.  The upshot is that there are about 700 opportunities for your event to happen (because the map $L(m)$ being largest prime factor of m is often injective, and in your case it won't be) below 2.5 times $10^{10}$, and that your event won't happen because the numbers involved are too close. (Specifically, $L(m)=L(n)=p$, and $m-n =kp$ where $L(k)$ is less than $p$ and usually less than 3, and in those cases $m/p$ and $n/p$ have sufficiently different sets of prime factors.). If I can achieve my aims while offloading data regarding your claim (e.g. a data file of the estimated 3000 $L$ pairs below $10^{12}$), I will do so and report back.  If you have several months of computer cycles to spare, I can provide a program so that you can join in the fun, AND get some data on your problem.

Gerhard "Another Opportunity For Communal Computing" Paseman, 2017.11.26.