What is the progress on the MIT LCS35 Time Capsule Crypto-Puzzle?
2) Has anyone claimed to make any progress with this challenge?


Ah that question I can answer now...

I found the solution on the 15th of April 2019 and sent it to the MIT's CSAIL on the 16th of April. Another team shall have the answer by the 11th or 12th of May (they used a FPGA).

I noticed around the end of 2015 that if I used GMP I could find the answer on my Core i7-6700 CPU in about three years and a half. So I basically let my computer open 24/7 so that it'd solve the "puzzle". 

The secret message and the two primes shall be announced on the 15th of May during a ceremony where the "time capsule" shall be opened.

Link to the CSAIL's article about it:

Programmers solve MIT’s 20-year-old cryptographic puzzle