Suppose I have a set of random vectors $f(a_1, \ldots, a_\ell) := (v_1, \ldots, v_m) \subset F_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a_i$'s are independent, uniformly distributed in $F_p \setminus \{0\}$, and each component of $f$ consists of some polynomial in the $a_i$'s whose coefficients are all in $\{0,1\}$, and such that the degree of each variable is at most $1$. For example, the following set of random vectors fits the discription:

$$ v_1 = (a_1, a_2 a_1, a_3); v_2 = (a_2, a_1 a_2 a_3 + a_2, 0); v_3 = (a_1 + a_2, a_2 + a_3, a_3 a_1)$$.

Now consider the quantity $\pi_p = \mathbb{P}(f(a_1, \ldots, a_\ell) \text{ spans } F_p^n)$. My question is, is $\pi_p$ monotone non-decreasing in $p$? If not can one give a counterexample? The motivation comes from a recent result of Yuval Peres and Allan Sly (Arxiv preprint arXiv:1105.4402, 201) giving the right order of mixing time for the most natural random walk on uni-upper triangular matrices over $F_p$. Knowing the above will extend their result of $\mathcal{O}(n^2)$ to $p$ that grows with $n$, which is highly anticipated.

Edit: Will Sawin below essentially solved an earlier version of this problem, where I forgot to state the degree condition on the $a_i$'s.