Find private key from knowing random exponents are an arithmetic series
We don't know if $s_1r_2−s_2r_1$ is coprime to $p−1$ so we can't immediately get $u$, but is it true that it would be coprime?


I don't believe coprimality is guarranteed, however we don't need it.

If they're not coprime, it means that there are multiple solutions to the linear equations we have; however:


The number of solutions are likely to be small; bounded by $\text{gcm}( s_1r_2−s_2r_1, p-1)$
We can list all those solutions efficiently


So, what we do is go through the list of possible $u, k$ solutions, plug each one into the $r_1 = g^k \pmod p$ relation, and see which one satisfies it; our work is done.