  Among the knots that are the connected sum of two torus knots with cobordism
distance 1, we characterize those that have 4-dimensional clasp number at least
2, and we show that their n-fold connected self-sum has 4-dimensional clasp
number at least 2n. Our proof works in the topological category. To contrast
this, we build a family of topologically slice knots for which the n-fold
connected self-sum has 4-ball genus n and 4-dimensional clasp number at least
2n.
