Polynomial van der Corput conjecture over function fields

Let ${\mathbb F_q}[t]$ be the polynomial ring over the finite field ${\mathbb F_q}$. For a set ${\mathcal H}\subset {\mathbb F_q}[t]\setminus\{0\}$, being van der Corput means that whenever a sequence $(a_x)_{x\in {\mathbb F_q}[t]}$ has equidistributed differences $(a_{x+h}-a_x)_{x\in {\mathbb F_q}[t]}$ for every $h\in {\mathcal H}$, the original sequence is equidistributed. A polynomial $\Phi(u)\in {\mathbb F_q}[t][u]\setminus\{0\}$ is said to have a root modulo $g$ if it has a root in the quotient ring modulo $g$. Polynomial van der Corput conjecture. If $\Phi$ has a root modulo every nonzero $g\in {\mathbb F_q}[t]$, then

$$\left\{\Phi(x):x\in {\mathbb F_q}[t]\right\}\setminus\{0\}$$

is a van der Corput set, and hence is intersective. This is the function-field analogue of the corresponding result for integer polynomials; the status evidence states that the conjecture remains open when the degree of $\Phi$ is at least $p$.