Monotonicity conjecture for the sequences $t_n$ and $T_n$

Let $(t_n)_{n=1}^{\infty}$ and $(T_n)_{n=1}^{\infty}$ be the sequences whose values are listed in the table preceding the conjecture. Monotonicity conjecture. For every $n\in\mathbb{Z}^+$,

$$t_{n+1}\leq t_n\quad\text{and}\quad T_n\leq T_{n+1}.$$

The conjecture asserts that $(t_n)$ is nonincreasing while $(T_n)$ is nondecreasing; the supplied source gives numerical evidence but no resolution.