A strict connection tree T of a graph G for a non-empty subset W ⊆ V (G), called terminal set, is a tree subgraph of G whose leaf set coincides with W . A non-terminal vertex v ∈ V (T ) \ W is called linker if its degree in T is exactly 2, and it is called router if its degree in T is at least 3. Given a graph G, a terminal set W ⊆ V (G) and two non-negative integers ℓ and r, the Strict terminal connection problem (S-TCP) asks whether G admits a strict connection tree for W with at most ℓ linkers and at most r routers. In the present extended abstract, we prove that S-TCP is NP-complete on chordal bipartite graphs even if ℓ is bounded by a constant.