The multiplicity estimate conjecture for Drinfeld quasi-modular forms

Let $q=p^e$ be a prime power, let $A=\mathbb{F}_q[\theta]$, and let $f$ be a non-zero Drinfeld quasi-modular form of weight $w$ and depth $l\geq 1$. Write $\nu_{\infty}(f)$ for its vanishing order at infinity. Multiplicity estimate conjecture. There exists a real number $c(q)>0$ such that

$$\nu_{\infty}(f)\leq c(q)\,l(w-l).$$

This conjecture concerns a uniform upper bound for the vanishing order in terms of the weight and depth. The paper proves a bound with an additional logarithmic factor, so the stated estimate remains open in the source.