Divisibility conjecture for the L-polynomials of the curves $E_k$

For a positive integer $k$, let $E_k$ be the smooth projective model over $\mathbb{F}_2$ of

$$y^2+xy=x^{2^k+3}+x.$$

Its L-polynomial is denoted by $\mathrm{L}_{E_k}(t)$. The $E_k$ divisibility conjecture. The L-polynomial of $E_k$ is divisible by the L-polynomial of $E_1$.

This is the second divisibility conjecture considered for a specific family of curves over $\mathbb{F}_2$. The supplied text does not state that it has been proved or disproved; the paper indicates that proving it requires methods different from those used for the $C_k$ family.