The diagonal-maximality conjecture for inverse Vandermonde matrices

Let $b>1$, and set

$$n_0=\left\lceil\log_b\left(1+\frac{1}{b}\right)\right\rceil.$$

For each $n$, let $M_b(n)$ denote the largest absolute value of an entry of the inverse Vandermonde matrix, and let $c_{i,j,n}$ denote its entries. Diagonal-maximality conjecture. For all sufficiently large $n$, there is some $i$ with $0\leq i\leq n_0$ such that

$$M_b(n)=|c_{i,i,n}|.$$

The preceding results show that the limit of $M_b(n)$ is attained asymptotically among finitely many entries, but do not establish that a diagonal entry is maximal for every sufficiently large $n$; this is the remaining unproved assertion.