Operator identity for convergent series

Let $T_i$ and $S_i$ be a sequence of bounded operators such that

$$\sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then a positive operator.

My question is now: Is it true or false that the following inequality holds

$$\sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k \ge \sum_{i=0}^{\infty} S_0^* T_i^* T_i S_0 ?$$

The only think that came to my mind were basic examples and manipulations, but they did not lead me anywhere.

Probably one could try to restrict to finite sums first.

But if all $T_0=I$ and all other $T_i=0$, $S_0=I$, $S_1=-I/2$, and all other $S_i=0$, then LHS$=I/4$, while RHS$=I$, violating the inequality.