The Chazon Ish is known to have opposed more than one man saying Kaddish at a time. In “Dinim Vehanhagos” he is quoted as comparing it to having two prayer leaders (“Shlia’ch tzibbur”)…
But did you know this was not an innovation by the Chazon Ish? This was, in fact, the ancient custom (not by Sefardim), as claimed here on the Treasures of Ashkenaz blog.
And I quote:
Only one person recites kaddish at a time (as opposed to the Sepharadic group kaddish).