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$10^{12}$ contains an injective divisor map, see On comparing two almost injective divisor maps for more detail. The upshot is that there are about 700 opportunities for your event to happen (because the map L(m)$L(m)$ being largest prime factor of m is often injective, and in your case it won't be) below 2.5 times 10^10$10^{10}$, and that your event won't happen because the numbers involved are too close. (Specifically, L(m)=L(n)=p$L(m)=L(n)=p$, and m-n =kp$m-n =kp$ where L(k)$L(k)$ is less than p$p$ and usually less than 3, and in those cases m/p$m/p$ and n/p$n/p$ 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$L$ pairs below 10^12$10^{12}$), 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.
