Is it true that at least two of any consecutive $2m$ positive integers cannot be divided by odd prime numbers less than $2m$?

To end this question, someone find a counterexample for $m=70$, where $$q\equiv -\lbrace 1,2,1,8,1,9,5,6,10,11,3,2,22,1,33,21,1,23,18,20,4,18,21,19,35,38,44,45,51,54,56,59,69 \rbrace \\ \mod {\lbrace 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109,113,127,131,137,139 \rbrace }.$$

This means that $q\equiv -1\pmod 3,q\equiv -2\pmod 5,\cdots,q\equiv -69\pmod {139}.$

A solution is $q=264782491305295395386123607229983302927523861123269753$ and every number in $q+1,\cdots,q+140$ can be divided by some odd prime number less that $140$.