Conjecture on rank-two tuples on the curve modulo p

Let $f\in\mathbb{Z}[x,y]$ and, for each prime $p$, let $C_p$ be the curve of points modulo $p$ defined by $f(x,y)\equiv 0\pmod p$. For a $k$-tuple $\mathbf{P}=((r_1,s_1),\ldots,(r_k,s_k))$ of element-wise distinct points on $C_p$, define

$$B_k(\mathbf{P})=\begin{bmatrix} f_x(r_1,s_1)&f_y(r_1,s_1)&f(r_1,s_1)/p\\ \vdots&\vdots&\vdots\\ f_x(r_k,s_k)&f_y(r_k,s_k)&f(r_k,s_k)/p\end{bmatrix}.$$

Let $m_{k,2}$ be the number of such tuples for which $B_k(\mathbf{P})$ has rank $2$ and its first two columns are linearly independent. Rank-two tuple conjecture. For all $k\geq 2$,

$$\lim_{p\to\infty}\frac{m_{k,2}}{p^2}=1.$$

This asymptotic is the key counting input for the paper's Poisson limit theorem: it describes the expected abundance of tuples satisfying the relevant rank condition, while the supplied text does not indicate whether the conjecture has been proved or disproved.