After Suicune has been caught in Tin Tower (which requires it to have been seen in Burned Tower and Ecruteak City, and for the player to have obtained seven badges and saved the Goldenrod City Radio Tower from Team Rocket), Eusine will appear in the Celadon City Pokémon Center and state that he is visiting his home town.
If Suicune, Entei and Raikou with the player's OT and ID appear in the player's party and/or the Pokémon Storage System (even if one of them is not in the current box), then Eusine will also mention that there are rumours of a rainbow-coloured Pokémon (Ho-Oh) appearing in Tin Tower and leave.
If a Suicune with the player's OT and ID is not present then Eusine won't say anything after his initial hometown message. If only an Entei and/or Raikou with the player's OT and ID are missing, then Eusine will ask the player if they have caught them.
Not having all three legendary beasts with the player's OT and ID is meant to force Eusine not to say anything after his initial message. However, due to a glitch, if the beast has a different OT only for characters six/seven, etc. (for example; CRYSTIE instead of CRYSTAL, then it will pass as being owned by the player and Eusine will mention the rumour about the rainbow colored Pokémon at Tin Tower and leave).