I'm wondering if there is any collision-resistant hash function $h^s(\cdot)$ satisfying that there is a fixed value $c$ such that, for each $s$, a value $x_s$ satisfying $h^s(x_s) = c$ is known. This would not contradict the collision-resistance property, nor preimage-resistance, but I have not been able to come up with any construction of this kind.
Does anyone know if this is possible, and if so, can point me to a particular construction?
I'm working on an exercise which asks to analyze the security of Merkle–Damgård transform when no $IV$ is used (or, equivalently, when it is set as the first block of the message). If a hash function like that I mention can be constructed, then I can build collisions on this construction.