Let $S$ and $T$ be sets of vectors from $\mathbb{R}^d$ such that $S$ and $T$ are at least different in one element.

Does there exist a random matrix $M \in \mathbb{R}^{d \times k}$, e.g., a gaussian matrix, such that the probability of $ \sum_{s \in S} s M = \sum_{t \in T} t M $ is small in terms of $k$?