Infinitude conjecture for even strong, weak and very weak alpha numbers

For real $x$, let $N_e(x,n)$, $\bar{N}_e(x,n)$ and $\bar{\bar{N}}_e(x,n)$ count respectively the strong, weak and very weak even alpha numbers $n$ not exceeding $x$. Even alpha-number infinitude conjecture. As $x\to\infty$,

$$N_e(x,n)\to\infty,\qquad \bar{N}_e(x,n)\to\infty,\qquad \bar{\bar{N}}_e(x,n)\to\infty.$$

This asserts that each of the three classes of even alpha numbers is infinite. The supplied text gives no resolution or further evidence for the conjecture.