Linear-average stopping-time conjecture for the polynomial Collatz map

Let $T\colon\mathbb{F}_2[x]\to\mathbb{F}_2[x]$ be the polynomial Collatz map, with $T(f)=(1+x)f+1$ for odd $f$ and $T(f)=f/x$ for even $f$. For polynomials of degree $d$, define the average stopping time by

$$\rho(d)=\frac{1}{2^d}\sum_{f\in\mathbb{F}_2[x],\,\operatorname{deg}(f)=d}t_{\rm \min}(f).$$

The linear-average stopping-time conjecture. The average stopping time $\rho(d)$ of the Collatz map $T$ on $\mathbb{F}_2[x]$ grows linearly in $d$. Experimental data suggests this behavior, but the source states that it has not been proved.