strike the previous comment, my proof was based on false assumption that any lattice is commeasurable with one which is proportional to unimodular

Yes, and this seems to give an answer indeed. I will post it in a couple of days, once I am entirely sure there are no errors

thanks! anyway, rank $\geq 6$ or $\geq 7$ is a usual assumption in these kind of applications

definitely! thanks for pointing this out, I amended the question

Sorry! I misstated the question: I should add an assumption that this 2-dimensional lattice is not definite!

I apologize for being overly blunt (and thanks to those who pointed this out). The question is edited. For the record, I said "I have read it when it first appeared in arxiv 10 years ago, there were not very subtle errors then. Is it still wrong? "