Positivity conjecture for coefficients of the Chebyshev matrix

Let $B_{\alpha,n}$ be the $n\times(2n-1)$ band matrix whose coefficients are

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

when $i\le k\le n-1+i$, and are zero otherwise. Let $B_\sigma(1)$ denote the associated coefficient indexed by the column selection $\sigma$. Chebyshev coefficient conjecture. For every $n\ge 2$, all coefficients $B_\sigma(1)$ are positive whenever $\alpha<1/n$. This conjecture would imply positivity of the derivative determinant and hence the proposed finite-order total positivity result. It is presented as open; the matrix is rectangular, which obstructs a straightforward induction.