Upper bound on the $2$-norm of inverse of a triangular Toeplitz matrix

Question

Consider a lower triangular square Toeplitz matrix

\begin{align} T_n = \begin{bmatrix} t_1 & 0 & 0 & \dots & 0\\ t_2 & t_1 & 0 & \dots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ t_n & t_{n-1} & t_{n-2} & \dots & t_1 \end{bmatrix} \end{align}

Matrix $T_n$ satisfies two conditions:

1. $t_1 > 0$
2. $\displaystyle\sum_{i=1}^n t_i^2 = c \leq 1$

Is it possible to find a lower bound on the smallest singular value of $T_n$ (or equivalently, an upper bound on $\|T_n^{-1}\|_2^2$) in terms of $t_1, c$ and $n$?

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Update: Based on the comment below, one can write $T^{-1}_n$ which is also triangular and Toeplitz. One upper bound on the 2-norm of Toeplitz is the following

$$\|T_n\|_2^2 \leq \sum_{i=1}^n t_i^2.$$

Is it possible to write a nice upper bound for the elements of $T^{-1}_n$?