  This paper presents evidence supporting the surprising conjecture that in the
topological category the slice genus of a satellite knot $P(K)$ is bounded
above by the sum of the slice genera of $K$ and $P(U)$. Our main result
establishes this conjecture for a variant of the topological slice genus, the
$\mathbb{Z}$-slice genus. As an application, we show that the $(n,1)$-cable of
any 3-genus 1 knot (e.g. the figure 8 or trefoil knot) has topological slice
genus at most 1. Further, we show that the lower bounds on the slice genus
coming from the Tristram-Levine and Casson-Gordon signatures cannot be used to
disprove the conjecture. Notably, the conjectured upper bound does not involve
the algebraic winding number of the pattern $P$. This stands in stark contrast
with the smooth category, where for example there are many genus 1 knots whose
$(n,1)$-cables have arbitrarily large smooth 4-genera.
