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.