The maximum number of dismal divisors in base b

Let $d_b(n)$ denote the number of dismal divisors of $n$ in base $b$, and let $n$ range over the $k$-digit base-$b$ numbers. Maximum-divisor conjecture. For every base $b\geq 3$, $d_b(n)$ has a unique maximum at

$$n=\frac{b^k-1}{b-1}=111\ldots1|_b.$$

The claim identifies the all-ones base-$b$ number as the unique $k$-digit number with the most dismal divisors. The paper gives this as a conjecture after noting that these numbers come close to the general upper bound, and does not report a resolution.