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.