Bozkurt's conjecture on mixed norms of Cauchy-Toeplitz matrices

Let $T_n$ be the Cauchy-Toeplitz matrix

$$T_n=\left[\frac{2}{1+2(i-j)}\right]_{i,j=1}^n.$$

For $1\leq p,q\leq\infty$, let $\|T_n\|_{p,q}$ denote its $\ell_{p,q}$ norm. Bozkurt's conjecture. The inequalities

$$n^{-\frac{1}{q}}\|T_n\|_{p,q}<4\left(\frac{1}{2}+\frac{1}{2^p-1}\right)^{\frac{1}{p}},\qquad p\geq q,$$

and

$$n^{-\frac{1}{q}}\|T_n\|_{p,q}\geq4\left(\frac{1}{2^p-1}\right)^{\frac{1}{p}},\qquad p<q,$$

are valid. The paper presents this as a conjecture proposed by D. Bozkurt and states in its abstract that it gives a complete answer to it.