Total positivity conjecture for the Chebyshev matrix

Let $B_{\alpha,n}$ be the rectangular band matrix defined by

$$B_{\alpha,n}(i,k)=\sum_{j=i}^n C_n^jC_n^{k-j}U^\alpha_{2j-k}$$

for $i\le k\le n-1+i$, with zero entries otherwise. A matrix is totally positive if all of its minors are non-negative. Chebyshev matrix conjecture. For every $n\ge 1$, the matrix $B_{\alpha,n}$ is totally positive whenever $\alpha<1/n$. This is stronger than positivity of the selected coefficients $B_\sigma(1)$ because it requires all minors to be non-negative. The paper verifies the claim for the first five values of $n$, but leaves the general case open.