The maximum number of dismal divisors in base 2

Let $d_2(n)$ denote the number of dismal divisors of the $k$-digit binary number $n$. Binary maximum-divisor conjecture. Among all $k$-digit numbers, the maximal value of $d_2(n)$ occurs at

$$n=2^k-2=111\ldots10|_2,$$

and this is the unique maximum except when $n=2$ or $n=4$. The conjecture is the binary counterpart of the preceding maximum-divisor claim, with explicit exceptional cases. The paper states it without a proof or resolution.