Linear-independence conjecture for the matrix associated with Paley-graph clique bounds

In the setting of the preceding discussion, let $M$ be the set of integers $m$ satisfying

$$\binom{n-1+\frac{q-1}{2}}{m}ot\equiv 0\pmod p,\qquad n\leq m\leq \frac{q-1}{2},$$

and let $A_{m,n}$ be the matrix defined in the paper, with $L(n)$ its associated index set. Matrix linear-independence conjecture. There is an integer $m\in M$ such that the last row of $A_{m,n}$ is linearly independent of the first $|L(n)|$ rows. The conjecture is proposed as a sufficient condition for proving the preceding improved bound on $\omega(P_q)$, but its status is not resolved in the supplied text.