Classification of depth-two zeta-like primitive tuples up to weight q^2

Let $q$ be a prime power, and consider primitive tuples of depth $2$ and weight at most $q^2$. Depth-two classification conjecture. The zeta-like primitive tuples are exactly

$$(i,j(q-1)),\qquad i=1,\dotsc,q,\quad j=i,\dotsc,\left\lfloor\frac{q^2-i}{q-1}\right\rfloor,$$

where the upper bound equals $q+1$ when $i<q$ and equals $q$ when $i=q$. The source notes that its theorems prove that these displayed tuples are zeta-like; the converse is conjectural.