The two largest dismal-divisor counts among odd binary numbers

Let $d_2(n)$ denote the number of dismal divisors of $n$, and let $n$ range over the odd $k$-digit binary numbers. Odd-binary divisor-count conjecture. The unique maximum of $d_2(n)$ occurs at

$$n=2^k-1=111\ldots111|_2.$$

If $k\geq 3$ and $k\neq 5$, the second-largest value occurs at

$$n=2^k-3=111\ldots101|_2,$$

and

$$n=2^k-2^{k-2}-1=101\ldots111|_2,$$

possibly together with other values of $n$. This refines the preceding binary maximum conjecture by restricting to odd numbers and specifying candidates for the second-largest value; the paper gives no proof or resolution.