Does the implication “security classic schnorr $=>$ multi-key security key prefixed schnorr” apply to short Schnorr signatures?

Question

My question refers to the paper "Multi-user Schnorr security, revisited" by Bernstein. Does the implication shown in the paper

security of classic Schnorr $\implies $ multi-key security of key-prefixed Schnorr

apply to short Schnorr signatures?

By short Schnorr signatures I mean those of size $3k$-bit that you get by truncating the hash output bits by $k$.